diff --git a/docs/mkdocs/docs/api/basic_json/number_float_t.md b/docs/mkdocs/docs/api/basic_json/number_float_t.md
index 83c7011c5..f6508a6c3 100644
--- a/docs/mkdocs/docs/api/basic_json/number_float_t.md
+++ b/docs/mkdocs/docs/api/basic_json/number_float_t.md
@@ -55,6 +55,10 @@ This implementation does exactly follow this approach, as it uses double precisi
 smaller than `-1.79769313486232e+308` and values greater than `1.79769313486232e+308` will be stored as NaN internally
 and be serialized to `null`.
 
+During deserialization (from JSON text or any of the binary formats), a finite number that does not fit into
+`number_float_t` is rejected with [`out_of_range.406`](../../home/exceptions.md#jsonexceptionout_of_range406), for
+example a double-precision number in a binary format when `number_float_t` is `#!cpp float`.
+
 #### Storage
 
 Floating-point number values are stored directly inside a `basic_json` type.
diff --git a/docs/mkdocs/docs/api/basic_json/number_integer_t.md b/docs/mkdocs/docs/api/basic_json/number_integer_t.md
index 9a2ffab7f..00f530d57 100644
--- a/docs/mkdocs/docs/api/basic_json/number_integer_t.md
+++ b/docs/mkdocs/docs/api/basic_json/number_integer_t.md
@@ -47,8 +47,9 @@ With the default values for `NumberIntegerType` (`std::int64_t`), the default va
 
 When the default type is used, the maximal integer number that can be stored is `9223372036854775807` (INT64_MAX) and
 the minimal integer number that can be stored is `-9223372036854775808` (INT64_MIN). Integer numbers that are out of
-range will yield over/underflow when used in a constructor. During deserialization, too large or small integer numbers
-will automatically be stored as [`number_unsigned_t`](number_unsigned_t.md) or [`number_float_t`](number_float_t.md).
+range will yield over/underflow when used in a constructor. During deserialization (from JSON text or any of the binary
+formats), too large or small integer numbers will automatically be stored as [`number_unsigned_t`](number_unsigned_t.md)
+or [`number_float_t`](number_float_t.md).
 
 [RFC 8259](https://tools.ietf.org/html/rfc8259) further states:
 > Note that when such software is used, numbers that are integers and are in the range $[-2^{53}+1, 2^{53}-1]$ are
diff --git a/docs/mkdocs/docs/api/basic_json/number_unsigned_t.md b/docs/mkdocs/docs/api/basic_json/number_unsigned_t.md
index 674f7711d..50ad2940b 100644
--- a/docs/mkdocs/docs/api/basic_json/number_unsigned_t.md
+++ b/docs/mkdocs/docs/api/basic_json/number_unsigned_t.md
@@ -48,8 +48,9 @@ With the default values for `NumberUnsignedType` (`std::uint64_t`), the default
 
 When the default type is used, the maximal integer number that can be stored is `18446744073709551615` (UINT64_MAX) and
 the minimal integer number that can be stored is `0`. Integer numbers that are out of range will yield over/underflow
-when used in a constructor. During deserialization, too large or small integer numbers will automatically be stored
-as [`number_integer_t`](number_integer_t.md) or [`number_float_t`](number_float_t.md).
+when used in a constructor. During deserialization (from JSON text or any of the binary formats), too large or small
+integer numbers will automatically be stored as [`number_integer_t`](number_integer_t.md) or
+[`number_float_t`](number_float_t.md).
 
 [RFC 8259](https://tools.ietf.org/html/rfc8259) further states:
 > Note that when such software is used, numbers that are integers and are in the range $[-2^{53}+1, 2^{53}-1]$ are
diff --git a/docs/mkdocs/docs/features/binary_formats/cbor.md b/docs/mkdocs/docs/features/binary_formats/cbor.md
index 8e6acf0fb..15d480ee1 100644
--- a/docs/mkdocs/docs/features/binary_formats/cbor.md
+++ b/docs/mkdocs/docs/features/binary_formats/cbor.md
@@ -168,9 +168,9 @@ The library maps CBOR types to JSON value types as follows:
 !!! warning "Negative integer overflow"
 
     CBOR negative integers (major type 1) are decoded as `-1 - n`. If the encoded magnitude `n` is too large for the
-    result to fit into `number_integer_t` (`std::int64_t` by default), parsing fails with a
-    [`parse_error.112`](../../home/exceptions.md#jsonexceptionparse_error112) exception rather than overflowing
-    silently.
+    result to fit into `number_integer_t` (`std::int64_t` by default), the result is stored as `number_float_t`, like
+    a too small integer in JSON text. For example, `-18446744073709551616` (`0x3B` followed by eight `0xFF` bytes) is
+    stored as `-1.8446744073709552e+19`.
 
 !!! warning "Object keys"
 
diff --git a/docs/mkdocs/docs/home/exceptions.md b/docs/mkdocs/docs/home/exceptions.md
index bf18baab1..9c1c4ba89 100644
--- a/docs/mkdocs/docs/home/exceptions.md
+++ b/docs/mkdocs/docs/home/exceptions.md
@@ -331,9 +331,6 @@ An unexpected byte was read in a [binary format](../features/binary_formats/inde
     [json.exception.parse_error.112] parse error at byte 15: syntax error while parsing BSON binary: byte array length cannot be negative, is -1
     ```
     ```
-    [json.exception.parse_error.112] parse error at byte 9: syntax error while parsing CBOR value: negative integer overflow
-    ```
-    ```
     [json.exception.parse_error.112] parse error at byte 5: syntax error while parsing BSON document: document size 6 does not match the number of bytes read (5)
     ```
 
@@ -847,13 +844,18 @@ The JSON Patch operations 'remove' and 'add' cannot be applied to the root eleme
 
 ### json.exception.out_of_range.406
 
-A parsed number could not be stored as without changing it to NaN or INF.
+A parsed number could not be stored without changing it to NaN or INF. For the binary formats, this happens when a
+finite floating-point number does not fit into [`number_float_t`](../api/basic_json/number_float_t.md), for example a
+double-precision number when `number_float_t` is `#!cpp float`.
 
-!!! failure "Example message"
+!!! failure "Example messages"
 
     ```
     number overflow parsing '10E1000'
     ```
+    ```
+    [json.exception.out_of_range.406] syntax error while parsing CBOR value: number overflow
+    ```
 
 ### json.exception.out_of_range.407
 
diff --git a/include/nlohmann/detail/input/binary_reader.hpp b/include/nlohmann/detail/input/binary_reader.hpp
index b0675c626..e3360aa97 100644
--- a/include/nlohmann/detail/input/binary_reader.hpp
+++ b/include/nlohmann/detail/input/binary_reader.hpp
@@ -559,7 +559,7 @@ class binary_reader
             case 0x01: // double
             {
                 double number{};
-                return get_number<double, true>(input_format_t::bson, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number<double, true>(input_format_t::bson, number) && emit_float(input_format_t::bson, number);
             }
 
             case 0x02: // string
@@ -600,19 +600,19 @@ class binary_reader
             case 0x10: // int32
             {
                 std::int32_t value{};
-                return get_number<std::int32_t, true>(input_format_t::bson, value) && sax->number_integer(value);
+                return get_number<std::int32_t, true>(input_format_t::bson, value) && emit_signed(value);
             }
 
             case 0x12: // int64
             {
                 std::int64_t value{};
-                return get_number<std::int64_t, true>(input_format_t::bson, value) && sax->number_integer(value);
+                return get_number<std::int64_t, true>(input_format_t::bson, value) && emit_signed(value);
             }
 
             case 0x11: // uint64
             {
                 std::uint64_t value{};
-                return get_number<std::uint64_t, true>(input_format_t::bson, value) && sax->number_unsigned(value);
+                return get_number<std::uint64_t, true>(input_format_t::bson, value) && emit_unsigned(value);
             }
 
             default: // anything else is not supported (yet)
@@ -638,14 +638,17 @@ class binary_reader
         {
             return false;
         }
-        const auto max_val = static_cast<NumberType>((std::numeric_limits<number_integer_t>::max)());
-        if (number > max_val)
+
+        // the value is -1 - number, which fits into number_integer_t
+        // whenever number does
+        if (JSON_HEDLEY_LIKELY(value_in_range_of<number_integer_t>(number)))
         {
-            return sax->parse_error(chars_read, get_token_string(),
-                                    parse_error::create(112, chars_read,
-                                            exception_message(input_format_t::cbor, "negative integer overflow", "value"), nullptr));
+            return sax->number_integer(static_cast<number_integer_t>(-1) - static_cast<number_integer_t>(number));
         }
-        return sax->number_integer(static_cast<number_integer_t>(-1) - static_cast<number_integer_t>(number));
+
+        // like the lexer does for JSON text, store a value too small for
+        // number_integer_t as number_float_t
+        return sax->number_float(static_cast<number_float_t>(-1) - static_cast<number_float_t>(number), "");
     }
 
     /*!
@@ -702,25 +705,25 @@ class binary_reader
             case 0x18: // Unsigned integer (one-byte uint8_t follows)
             {
                 std::uint8_t number{};
-                return get_number(input_format_t::cbor, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::cbor, number) && emit_unsigned(number);
             }
 
             case 0x19: // Unsigned integer (two-byte uint16_t follows)
             {
                 std::uint16_t number{};
-                return get_number(input_format_t::cbor, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::cbor, number) && emit_unsigned(number);
             }
 
             case 0x1A: // Unsigned integer (four-byte uint32_t follows)
             {
                 std::uint32_t number{};
-                return get_number(input_format_t::cbor, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::cbor, number) && emit_unsigned(number);
             }
 
             case 0x1B: // Unsigned integer (eight-byte uint64_t follows)
             {
                 std::uint64_t number{};
-                return get_number(input_format_t::cbor, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::cbor, number) && emit_unsigned(number);
             }
 
             // Negative integer -1-0x00..-1-0x17 (-1..-24)
@@ -1165,13 +1168,13 @@ class binary_reader
             case 0xFA: // Single-Precision Float (four-byte IEEE 754)
             {
                 float number{};
-                return get_number(input_format_t::cbor, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::cbor, number) && emit_float(input_format_t::cbor, number);
             }
 
             case 0xFB: // Double-Precision Float (eight-byte IEEE 754)
             {
                 double number{};
-                return get_number(input_format_t::cbor, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::cbor, number) && emit_float(input_format_t::cbor, number);
             }
 
             default: // anything else (0xFF is handled inside the other types)
@@ -1861,61 +1864,61 @@ class binary_reader
             case 0xCA: // float 32
             {
                 float number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::msgpack, number) && emit_float(input_format_t::msgpack, number);
             }
 
             case 0xCB: // float 64
             {
                 double number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::msgpack, number) && emit_float(input_format_t::msgpack, number);
             }
 
             case 0xCC: // uint 8
             {
                 std::uint8_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::msgpack, number) && emit_unsigned(number);
             }
 
             case 0xCD: // uint 16
             {
                 std::uint16_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::msgpack, number) && emit_unsigned(number);
             }
 
             case 0xCE: // uint 32
             {
                 std::uint32_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::msgpack, number) && emit_unsigned(number);
             }
 
             case 0xCF: // uint 64
             {
                 std::uint64_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::msgpack, number) && emit_unsigned(number);
             }
 
             case 0xD0: // int 8
             {
                 std::int8_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_integer(number);
+                return get_number(input_format_t::msgpack, number) && emit_signed(number);
             }
 
             case 0xD1: // int 16
             {
                 std::int16_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_integer(number);
+                return get_number(input_format_t::msgpack, number) && emit_signed(number);
             }
 
             case 0xD2: // int 32
             {
                 std::int32_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_integer(number);
+                return get_number(input_format_t::msgpack, number) && emit_signed(number);
             }
 
             case 0xD3: // int 64
             {
                 std::int64_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_integer(number);
+                return get_number(input_format_t::msgpack, number) && emit_signed(number);
             }
 
             case 0xDC: // array 16
@@ -2756,7 +2759,7 @@ class binary_reader
                         {
                             return sax->parse_error(chars_read, get_token_string(), out_of_range::create(408, exception_message(input_format, "excessive ndarray size caused overflow", "size"), nullptr));
                         }
-                        if (JSON_HEDLEY_UNLIKELY(!sax->number_unsigned(static_cast<number_unsigned_t>(i))))
+                        if (JSON_HEDLEY_UNLIKELY(!emit_unsigned(i)))
                         {
                             return false;
                         }
@@ -2888,37 +2891,37 @@ class binary_reader
                     break;
                 }
                 std::uint8_t number{};
-                return get_number(input_format, number) && sax->number_unsigned(number);
+                return get_number(input_format, number) && emit_unsigned(number);
             }
 
             case 'U':
             {
                 std::uint8_t number{};
-                return get_number(input_format, number) && sax->number_unsigned(number);
+                return get_number(input_format, number) && emit_unsigned(number);
             }
 
             case 'i':
             {
                 std::int8_t number{};
-                return get_number(input_format, number) && sax->number_integer(number);
+                return get_number(input_format, number) && emit_signed(number);
             }
 
             case 'I':
             {
                 std::int16_t number{};
-                return get_number(input_format, number) && sax->number_integer(number);
+                return get_number(input_format, number) && emit_signed(number);
             }
 
             case 'l':
             {
                 std::int32_t number{};
-                return get_number(input_format, number) && sax->number_integer(number);
+                return get_number(input_format, number) && emit_signed(number);
             }
 
             case 'L':
             {
                 std::int64_t number{};
-                return get_number(input_format, number) && sax->number_integer(number);
+                return get_number(input_format, number) && emit_signed(number);
             }
 
             case 'u':
@@ -2928,7 +2931,7 @@ class binary_reader
                     break;
                 }
                 std::uint16_t number{};
-                return get_number(input_format, number) && sax->number_unsigned(number);
+                return get_number(input_format, number) && emit_unsigned(number);
             }
 
             case 'm':
@@ -2938,7 +2941,7 @@ class binary_reader
                     break;
                 }
                 std::uint32_t number{};
-                return get_number(input_format, number) && sax->number_unsigned(number);
+                return get_number(input_format, number) && emit_unsigned(number);
             }
 
             case 'M':
@@ -2948,7 +2951,7 @@ class binary_reader
                     break;
                 }
                 std::uint64_t number{};
-                return get_number(input_format, number) && sax->number_unsigned(number);
+                return get_number(input_format, number) && emit_unsigned(number);
             }
 
             case 'h':
@@ -3006,13 +3009,13 @@ class binary_reader
             case 'd':
             {
                 float number{};
-                return get_number(input_format, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format, number) && emit_float(input_format, number);
             }
 
             case 'D':
             {
                 double number{};
-                return get_number(input_format, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format, number) && emit_float(input_format, number);
             }
 
             case 'H':
@@ -3479,13 +3482,13 @@ class binary_reader
             case 0x8E: // binary32
             {
                 float number{};
-                return get_number(input_format_t::bon8, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::bon8, number) && emit_float(input_format_t::bon8, number);
             }
 
             case 0x8F: // binary64
             {
                 double number{};
-                return get_number(input_format_t::bon8, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::bon8, number) && emit_float(input_format_t::bon8, number);
             }
 
             case 0xF8:
@@ -3551,7 +3554,9 @@ class binary_reader
     @brief pass an integer to the SAX parser
 
     Non-negative integers are passed as unsigned, negative integers as signed
-    numbers, like the other binary formats do.
+    numbers, like the other binary formats do. A value that does not fit the
+    number type is passed as described for @ref emit_unsigned and
+    @ref emit_signed.
 
     @param[in] number  the integer
     @return whether the SAX parser accepted the value
@@ -3560,9 +3565,9 @@ class binary_reader
     {
         if (number >= 0)
         {
-            return sax->number_unsigned(static_cast<number_unsigned_t>(number));
+            return emit_unsigned(static_cast<std::uint64_t>(number));
         }
-        return sax->number_integer(static_cast<number_integer_t>(number));
+        return emit_signed(number);
     }
 
     /*!
@@ -3619,8 +3624,7 @@ class binary_reader
             value = (value << 8) | static_cast<std::int64_t>(current);
         }
 
-        return negative ? sax->number_integer(static_cast<number_integer_t>(-(value + offset)))
-               : sax->number_unsigned(static_cast<number_unsigned_t>(value + offset));
+        return emit_bon8_integer(negative ? -(value + offset) : value + offset);
     }
 
     /*!
@@ -3917,6 +3921,79 @@ class binary_reader
         return true;
     }
 
+    /*!
+    @brief pass a signed integer read from the input to the SAX parser
+
+    Like the lexer does for JSON text, a value that does not fit into
+    number_integer_t is passed as number_unsigned_t if it is non-negative and
+    fits there, and as number_float_t otherwise. With the default number
+    types, every integer the binary formats can encode fits, so this only
+    matters for narrower custom number types.
+
+    @tparam NumberType a signed integer type
+    @param[in] number  the integer
+    @return whether the SAX parser accepted the value
+    */
+    template<typename NumberType>
+    bool emit_signed(const NumberType number)
+    {
+        if (JSON_HEDLEY_LIKELY(value_in_range_of<number_integer_t>(number)))
+        {
+            return sax->number_integer(static_cast<number_integer_t>(number));
+        }
+        if (value_in_range_of<number_unsigned_t>(number))
+        {
+            return sax->number_unsigned(static_cast<number_unsigned_t>(number));
+        }
+        return sax->number_float(static_cast<number_float_t>(number), "");
+    }
+
+    /*!
+    @brief pass an unsigned integer read from the input to the SAX parser
+
+    Like the lexer does for JSON text, a value that does not fit into
+    number_unsigned_t is passed as number_float_t.
+
+    @tparam NumberType an unsigned integer type
+    @param[in] number  the integer
+    @return whether the SAX parser accepted the value
+    */
+    template<typename NumberType>
+    bool emit_unsigned(const NumberType number)
+    {
+        if (JSON_HEDLEY_LIKELY(value_in_range_of<number_unsigned_t>(number)))
+        {
+            return sax->number_unsigned(static_cast<number_unsigned_t>(number));
+        }
+        return sax->number_float(static_cast<number_float_t>(number), "");
+    }
+
+    /*!
+    @brief pass a floating-point number read from the input to the SAX parser
+
+    Like the lexer does for JSON text, a finite value that overflows
+    number_float_t is rejected instead of silently becoming infinity. Infinity
+    and NaN in the input are passed on unchanged.
+
+    @tparam NumberType a floating-point type
+    @param[in] format  the current format (for diagnostics)
+    @param[in] number  the number
+    @return whether the SAX parser accepted the value
+
+    @throw out_of_range.406 if a finite @a number overflows number_float_t
+    */
+    template<typename NumberType>
+    bool emit_float(const input_format_t format, const NumberType number)
+    {
+        const auto result = static_cast<number_float_t>(number);
+        if (JSON_HEDLEY_UNLIKELY(std::isfinite(number) && !std::isfinite(result)))
+        {
+            return sax->parse_error(chars_read, get_token_string(),
+                                    out_of_range::create(406, exception_message(format, "number overflow", "value"), nullptr));
+        }
+        return sax->number_float(result, "");
+    }
+
     /*!
     @brief create a string by reading characters from the input
 
diff --git a/single_include/nlohmann/json.hpp b/single_include/nlohmann/json.hpp
index 0e3cae486..152e6c511 100644
--- a/single_include/nlohmann/json.hpp
+++ b/single_include/nlohmann/json.hpp
@@ -13294,7 +13294,7 @@ class binary_reader
             case 0x01: // double
             {
                 double number{};
-                return get_number<double, true>(input_format_t::bson, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number<double, true>(input_format_t::bson, number) && emit_float(input_format_t::bson, number);
             }
 
             case 0x02: // string
@@ -13335,19 +13335,19 @@ class binary_reader
             case 0x10: // int32
             {
                 std::int32_t value{};
-                return get_number<std::int32_t, true>(input_format_t::bson, value) && sax->number_integer(value);
+                return get_number<std::int32_t, true>(input_format_t::bson, value) && emit_signed(value);
             }
 
             case 0x12: // int64
             {
                 std::int64_t value{};
-                return get_number<std::int64_t, true>(input_format_t::bson, value) && sax->number_integer(value);
+                return get_number<std::int64_t, true>(input_format_t::bson, value) && emit_signed(value);
             }
 
             case 0x11: // uint64
             {
                 std::uint64_t value{};
-                return get_number<std::uint64_t, true>(input_format_t::bson, value) && sax->number_unsigned(value);
+                return get_number<std::uint64_t, true>(input_format_t::bson, value) && emit_unsigned(value);
             }
 
             default: // anything else is not supported (yet)
@@ -13373,14 +13373,17 @@ class binary_reader
         {
             return false;
         }
-        const auto max_val = static_cast<NumberType>((std::numeric_limits<number_integer_t>::max)());
-        if (number > max_val)
+
+        // the value is -1 - number, which fits into number_integer_t
+        // whenever number does
+        if (JSON_HEDLEY_LIKELY(value_in_range_of<number_integer_t>(number)))
         {
-            return sax->parse_error(chars_read, get_token_string(),
-                                    parse_error::create(112, chars_read,
-                                            exception_message(input_format_t::cbor, "negative integer overflow", "value"), nullptr));
+            return sax->number_integer(static_cast<number_integer_t>(-1) - static_cast<number_integer_t>(number));
         }
-        return sax->number_integer(static_cast<number_integer_t>(-1) - static_cast<number_integer_t>(number));
+
+        // like the lexer does for JSON text, store a value too small for
+        // number_integer_t as number_float_t
+        return sax->number_float(static_cast<number_float_t>(-1) - static_cast<number_float_t>(number), "");
     }
 
     /*!
@@ -13437,25 +13440,25 @@ class binary_reader
             case 0x18: // Unsigned integer (one-byte uint8_t follows)
             {
                 std::uint8_t number{};
-                return get_number(input_format_t::cbor, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::cbor, number) && emit_unsigned(number);
             }
 
             case 0x19: // Unsigned integer (two-byte uint16_t follows)
             {
                 std::uint16_t number{};
-                return get_number(input_format_t::cbor, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::cbor, number) && emit_unsigned(number);
             }
 
             case 0x1A: // Unsigned integer (four-byte uint32_t follows)
             {
                 std::uint32_t number{};
-                return get_number(input_format_t::cbor, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::cbor, number) && emit_unsigned(number);
             }
 
             case 0x1B: // Unsigned integer (eight-byte uint64_t follows)
             {
                 std::uint64_t number{};
-                return get_number(input_format_t::cbor, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::cbor, number) && emit_unsigned(number);
             }
 
             // Negative integer -1-0x00..-1-0x17 (-1..-24)
@@ -13900,13 +13903,13 @@ class binary_reader
             case 0xFA: // Single-Precision Float (four-byte IEEE 754)
             {
                 float number{};
-                return get_number(input_format_t::cbor, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::cbor, number) && emit_float(input_format_t::cbor, number);
             }
 
             case 0xFB: // Double-Precision Float (eight-byte IEEE 754)
             {
                 double number{};
-                return get_number(input_format_t::cbor, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::cbor, number) && emit_float(input_format_t::cbor, number);
             }
 
             default: // anything else (0xFF is handled inside the other types)
@@ -14596,61 +14599,61 @@ class binary_reader
             case 0xCA: // float 32
             {
                 float number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::msgpack, number) && emit_float(input_format_t::msgpack, number);
             }
 
             case 0xCB: // float 64
             {
                 double number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::msgpack, number) && emit_float(input_format_t::msgpack, number);
             }
 
             case 0xCC: // uint 8
             {
                 std::uint8_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::msgpack, number) && emit_unsigned(number);
             }
 
             case 0xCD: // uint 16
             {
                 std::uint16_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::msgpack, number) && emit_unsigned(number);
             }
 
             case 0xCE: // uint 32
             {
                 std::uint32_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::msgpack, number) && emit_unsigned(number);
             }
 
             case 0xCF: // uint 64
             {
                 std::uint64_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_unsigned(number);
+                return get_number(input_format_t::msgpack, number) && emit_unsigned(number);
             }
 
             case 0xD0: // int 8
             {
                 std::int8_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_integer(number);
+                return get_number(input_format_t::msgpack, number) && emit_signed(number);
             }
 
             case 0xD1: // int 16
             {
                 std::int16_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_integer(number);
+                return get_number(input_format_t::msgpack, number) && emit_signed(number);
             }
 
             case 0xD2: // int 32
             {
                 std::int32_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_integer(number);
+                return get_number(input_format_t::msgpack, number) && emit_signed(number);
             }
 
             case 0xD3: // int 64
             {
                 std::int64_t number{};
-                return get_number(input_format_t::msgpack, number) && sax->number_integer(number);
+                return get_number(input_format_t::msgpack, number) && emit_signed(number);
             }
 
             case 0xDC: // array 16
@@ -15491,7 +15494,7 @@ class binary_reader
                         {
                             return sax->parse_error(chars_read, get_token_string(), out_of_range::create(408, exception_message(input_format, "excessive ndarray size caused overflow", "size"), nullptr));
                         }
-                        if (JSON_HEDLEY_UNLIKELY(!sax->number_unsigned(static_cast<number_unsigned_t>(i))))
+                        if (JSON_HEDLEY_UNLIKELY(!emit_unsigned(i)))
                         {
                             return false;
                         }
@@ -15623,37 +15626,37 @@ class binary_reader
                     break;
                 }
                 std::uint8_t number{};
-                return get_number(input_format, number) && sax->number_unsigned(number);
+                return get_number(input_format, number) && emit_unsigned(number);
             }
 
             case 'U':
             {
                 std::uint8_t number{};
-                return get_number(input_format, number) && sax->number_unsigned(number);
+                return get_number(input_format, number) && emit_unsigned(number);
             }
 
             case 'i':
             {
                 std::int8_t number{};
-                return get_number(input_format, number) && sax->number_integer(number);
+                return get_number(input_format, number) && emit_signed(number);
             }
 
             case 'I':
             {
                 std::int16_t number{};
-                return get_number(input_format, number) && sax->number_integer(number);
+                return get_number(input_format, number) && emit_signed(number);
             }
 
             case 'l':
             {
                 std::int32_t number{};
-                return get_number(input_format, number) && sax->number_integer(number);
+                return get_number(input_format, number) && emit_signed(number);
             }
 
             case 'L':
             {
                 std::int64_t number{};
-                return get_number(input_format, number) && sax->number_integer(number);
+                return get_number(input_format, number) && emit_signed(number);
             }
 
             case 'u':
@@ -15663,7 +15666,7 @@ class binary_reader
                     break;
                 }
                 std::uint16_t number{};
-                return get_number(input_format, number) && sax->number_unsigned(number);
+                return get_number(input_format, number) && emit_unsigned(number);
             }
 
             case 'm':
@@ -15673,7 +15676,7 @@ class binary_reader
                     break;
                 }
                 std::uint32_t number{};
-                return get_number(input_format, number) && sax->number_unsigned(number);
+                return get_number(input_format, number) && emit_unsigned(number);
             }
 
             case 'M':
@@ -15683,7 +15686,7 @@ class binary_reader
                     break;
                 }
                 std::uint64_t number{};
-                return get_number(input_format, number) && sax->number_unsigned(number);
+                return get_number(input_format, number) && emit_unsigned(number);
             }
 
             case 'h':
@@ -15741,13 +15744,13 @@ class binary_reader
             case 'd':
             {
                 float number{};
-                return get_number(input_format, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format, number) && emit_float(input_format, number);
             }
 
             case 'D':
             {
                 double number{};
-                return get_number(input_format, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format, number) && emit_float(input_format, number);
             }
 
             case 'H':
@@ -16214,13 +16217,13 @@ class binary_reader
             case 0x8E: // binary32
             {
                 float number{};
-                return get_number(input_format_t::bon8, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::bon8, number) && emit_float(input_format_t::bon8, number);
             }
 
             case 0x8F: // binary64
             {
                 double number{};
-                return get_number(input_format_t::bon8, number) && sax->number_float(static_cast<number_float_t>(number), "");
+                return get_number(input_format_t::bon8, number) && emit_float(input_format_t::bon8, number);
             }
 
             case 0xF8:
@@ -16286,7 +16289,9 @@ class binary_reader
     @brief pass an integer to the SAX parser
 
     Non-negative integers are passed as unsigned, negative integers as signed
-    numbers, like the other binary formats do.
+    numbers, like the other binary formats do. A value that does not fit the
+    number type is passed as described for @ref emit_unsigned and
+    @ref emit_signed.
 
     @param[in] number  the integer
     @return whether the SAX parser accepted the value
@@ -16295,9 +16300,9 @@ class binary_reader
     {
         if (number >= 0)
         {
-            return sax->number_unsigned(static_cast<number_unsigned_t>(number));
+            return emit_unsigned(static_cast<std::uint64_t>(number));
         }
-        return sax->number_integer(static_cast<number_integer_t>(number));
+        return emit_signed(number);
     }
 
     /*!
@@ -16354,8 +16359,7 @@ class binary_reader
             value = (value << 8) | static_cast<std::int64_t>(current);
         }
 
-        return negative ? sax->number_integer(static_cast<number_integer_t>(-(value + offset)))
-               : sax->number_unsigned(static_cast<number_unsigned_t>(value + offset));
+        return emit_bon8_integer(negative ? -(value + offset) : value + offset);
     }
 
     /*!
@@ -16652,6 +16656,79 @@ class binary_reader
         return true;
     }
 
+    /*!
+    @brief pass a signed integer read from the input to the SAX parser
+
+    Like the lexer does for JSON text, a value that does not fit into
+    number_integer_t is passed as number_unsigned_t if it is non-negative and
+    fits there, and as number_float_t otherwise. With the default number
+    types, every integer the binary formats can encode fits, so this only
+    matters for narrower custom number types.
+
+    @tparam NumberType a signed integer type
+    @param[in] number  the integer
+    @return whether the SAX parser accepted the value
+    */
+    template<typename NumberType>
+    bool emit_signed(const NumberType number)
+    {
+        if (JSON_HEDLEY_LIKELY(value_in_range_of<number_integer_t>(number)))
+        {
+            return sax->number_integer(static_cast<number_integer_t>(number));
+        }
+        if (value_in_range_of<number_unsigned_t>(number))
+        {
+            return sax->number_unsigned(static_cast<number_unsigned_t>(number));
+        }
+        return sax->number_float(static_cast<number_float_t>(number), "");
+    }
+
+    /*!
+    @brief pass an unsigned integer read from the input to the SAX parser
+
+    Like the lexer does for JSON text, a value that does not fit into
+    number_unsigned_t is passed as number_float_t.
+
+    @tparam NumberType an unsigned integer type
+    @param[in] number  the integer
+    @return whether the SAX parser accepted the value
+    */
+    template<typename NumberType>
+    bool emit_unsigned(const NumberType number)
+    {
+        if (JSON_HEDLEY_LIKELY(value_in_range_of<number_unsigned_t>(number)))
+        {
+            return sax->number_unsigned(static_cast<number_unsigned_t>(number));
+        }
+        return sax->number_float(static_cast<number_float_t>(number), "");
+    }
+
+    /*!
+    @brief pass a floating-point number read from the input to the SAX parser
+
+    Like the lexer does for JSON text, a finite value that overflows
+    number_float_t is rejected instead of silently becoming infinity. Infinity
+    and NaN in the input are passed on unchanged.
+
+    @tparam NumberType a floating-point type
+    @param[in] format  the current format (for diagnostics)
+    @param[in] number  the number
+    @return whether the SAX parser accepted the value
+
+    @throw out_of_range.406 if a finite @a number overflows number_float_t
+    */
+    template<typename NumberType>
+    bool emit_float(const input_format_t format, const NumberType number)
+    {
+        const auto result = static_cast<number_float_t>(number);
+        if (JSON_HEDLEY_UNLIKELY(std::isfinite(number) && !std::isfinite(result)))
+        {
+            return sax->parse_error(chars_read, get_token_string(),
+                                    out_of_range::create(406, exception_message(format, "number overflow", "value"), nullptr));
+        }
+        return sax->number_float(result, "");
+    }
+
     /*!
     @brief create a string by reading characters from the input
 
diff --git a/tests/src/unit-binary_formats.cpp b/tests/src/unit-binary_formats.cpp
index ed6d89911..f1fb06582 100644
--- a/tests/src/unit-binary_formats.cpp
+++ b/tests/src/unit-binary_formats.cpp
@@ -11,7 +11,12 @@
 #include <nlohmann/json.hpp>
 using nlohmann::json;
 
+#include <cmath>
 #include <fstream>
+#include <limits>
+#include <map>
+#include <string>
+#include <vector>
 #include "make_test_data_available.hpp"
 
 TEST_CASE("Binary Formats" * doctest::skip())
@@ -224,3 +229,114 @@ TEST_CASE("Binary Formats" * doctest::skip())
         CHECK((100.0 * double(ubjson_3_size) / double(json_size)) == Approx(89.450));
     }
 }
+
+TEST_CASE("Binary formats with narrow number types")
+{
+    // Numbers that do not fit the number types are handled like the lexer
+    // handles them in JSON text: an integer that fits neither integer type is
+    // stored as a floating-point number, and a finite floating-point number
+    // that overflows number_float_t is rejected with out_of_range.406.
+    using narrow_json = nlohmann::basic_json<std::map, std::vector, std::string, bool, std::int32_t, std::uint32_t, float>;
+    using bytes = std::vector<std::uint8_t>;
+
+    struct binary_format
+    {
+        const char* name;
+        bytes (*encode)(const json&);
+        narrow_json (*decode)(const bytes&, bool);
+    };
+
+    const std::vector<binary_format> formats =
+    {
+        {
+            "CBOR", [](const json & j) { return json::to_cbor(j); },
+            [](const bytes & v, bool allow_exceptions)
+            {
+                return narrow_json::from_cbor(v, true, allow_exceptions);
+            }
+        },
+        {
+            "MessagePack", [](const json & j) { return json::to_msgpack(j); },
+            [](const bytes & v, bool allow_exceptions)
+            {
+                return narrow_json::from_msgpack(v, true, allow_exceptions);
+            }
+        },
+        {
+            "UBJSON", [](const json & j) { return json::to_ubjson(j); },
+            [](const bytes & v, bool allow_exceptions)
+            {
+                return narrow_json::from_ubjson(v, true, allow_exceptions);
+            }
+        },
+        {
+            "BJData", [](const json & j) { return json::to_bjdata(j); },
+            [](const bytes & v, bool allow_exceptions)
+            {
+                return narrow_json::from_bjdata(v, true, allow_exceptions);
+            }
+        },
+        {
+            // BSON can only store numbers as object members
+            "BSON", [](const json & j) { return json::to_bson(json{{"a", j}}); },
+            [](const bytes & v, bool allow_exceptions)
+            {
+                const auto result = narrow_json::from_bson(v, true, allow_exceptions);
+                return result.is_discarded() ? result : result.at("a");
+            }
+        },
+        {
+            "BON8", [](const json & j) { return json::to_bon8(j); },
+            [](const bytes & v, bool allow_exceptions)
+            {
+                return narrow_json::from_bon8(v, true, allow_exceptions);
+            }
+        },
+    };
+
+    for (const auto& format : formats)
+    {
+        const std::string name = format.name;
+        INFO("format := ", name);
+        const auto roundtrip = [&format](const json & j)
+        {
+            return format.decode(format.encode(j), true);
+        };
+
+        // integers that fit keep their type
+        CHECK(roundtrip(json(-5)).is_number_integer());
+        CHECK(roundtrip(json(-5)).get<std::int32_t>() == -5);
+        CHECK(roundtrip(json(3000000000u)).is_number_unsigned());
+        CHECK(roundtrip(json(3000000000u)).get<std::uint32_t>() == 3000000000u);
+
+        // integers that fit neither integer type are stored as float
+        CHECK(roundtrip(json(5000000000u)).is_number_float());
+        CHECK(roundtrip(json(5000000000u)).get<float>() == 5000000000.0f);
+        if (name != "BON8") // BON8 cannot encode integers above INT64_MAX
+        {
+            CHECK(roundtrip(json(10000000000000000000u)).is_number_float());
+            CHECK(roundtrip(json(10000000000000000000u)).get<float>() == 10000000000000000000.0f);
+        }
+        CHECK(roundtrip(json(-3000000000)).is_number_float());
+        CHECK(roundtrip(json(-3000000000)).get<float>() == -3000000000.0f);
+        CHECK(roundtrip(json(-5000000000)).is_number_float());
+        CHECK(roundtrip(json(-5000000000)).get<float>() == -5000000000.0f);
+
+        // floating-point numbers that fit
+        CHECK(roundtrip(json(1.5)).get<float>() == 1.5f);
+        const auto just_above_max = std::nextafter(static_cast<double>((std::numeric_limits<float>::max)()),
+                                    std::numeric_limits<double>::infinity());
+        CHECK(roundtrip(json(just_above_max)).get<float>() == (std::numeric_limits<float>::max)());
+
+        // infinity and NaN are passed on
+        CHECK(std::isinf(roundtrip(json(std::numeric_limits<double>::infinity())).get<float>()));
+        CHECK(std::isnan(roundtrip(json(std::numeric_limits<double>::quiet_NaN())).get<float>()));
+
+        // finite floating-point numbers that overflow number_float_t are rejected
+        const std::string message = "[json.exception.out_of_range.406] syntax error while parsing " + name
+                                    + " value: number overflow";
+        CHECK_THROWS_WITH_AS(roundtrip(json(1e300)), message.c_str(), narrow_json::out_of_range&);
+        CHECK_THROWS_WITH_AS(roundtrip(json(-1e300)), message.c_str(), narrow_json::out_of_range&);
+        CHECK(format.decode(format.encode(json(1e300)), false).is_discarded());
+    }
+}
diff --git a/tests/src/unit-cbor.cpp b/tests/src/unit-cbor.cpp
index 6bd792f8a..fe81135d5 100644
--- a/tests/src/unit-cbor.cpp
+++ b/tests/src/unit-cbor.cpp
@@ -3146,7 +3146,8 @@ TEST_CASE("Tagged values")
         // CBOR encodes negative integers as: result = -1 - n
         // For type 0x3B, n is an 8-byte uint64_t. Valid range for n with
         // the default int64_t is [0, INT64_MAX], producing results in [INT64_MIN, -1].
-        // When n > INT64_MAX, the result exceeds int64_t range and is rejected.
+        // When n > INT64_MAX, the result exceeds int64_t range and is stored
+        // as a floating-point number, as the lexer does for JSON text.
 
         SECTION("n = 0 is valid (result = -1)")
         {
@@ -3167,33 +3168,34 @@ TEST_CASE("Tagged values")
             CHECK(result.get<int64_t>() == (std::numeric_limits<int64_t>::min)());
         }
 
-        SECTION("n = INT64_MAX + 1 is rejected (overflow)")
+        SECTION("n = INT64_MAX + 1 is stored as float")
         {
             // n = INT64_MAX + 1 (0x8000000000000000)
-            // result = -1 - n = -9223372036854775809, which exceeds int64_t range
+            // result = -1 - n = -9223372036854775809, which exceeds int64_t range;
+            // the nearest double is -9223372036854775808.0
             const std::vector<uint8_t> input = {0x3B, 0x80, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00};
-            json _;
-            CHECK_THROWS_WITH_AS(_ = json::from_cbor(input),
-                                 "[json.exception.parse_error.112] parse error at byte 9: syntax error while parsing CBOR value: negative integer overflow",
-                                 json::parse_error);
+            const auto result = json::from_cbor(input);
+            CHECK(result.is_number_float());
+            CHECK(result.get<double>() == -9223372036854775808.0);
+            CHECK(result == json::parse("-9223372036854775809"));
         }
 
-        SECTION("n = UINT64_MAX is rejected (overflow)")
+        SECTION("n = UINT64_MAX is stored as float")
         {
             // n = UINT64_MAX (0xFFFFFFFFFFFFFFFF)
             // result = -1 - n = -18446744073709551616, which exceeds int64_t range
             const std::vector<uint8_t> input = {0x3B, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF};
-            json _;
-            CHECK_THROWS_WITH_AS(_ = json::from_cbor(input),
-                                 "[json.exception.parse_error.112] parse error at byte 9: syntax error while parsing CBOR value: negative integer overflow",
-                                 json::parse_error);
+            const auto result = json::from_cbor(input);
+            CHECK(result.is_number_float());
+            CHECK(result.get<double>() == -18446744073709551616.0);
+            CHECK(result == json::parse("-18446744073709551616"));
         }
 
-        SECTION("overflow with allow_exceptions=false returns discarded")
+        SECTION("overflow with allow_exceptions=false is not an error")
         {
             const std::vector<uint8_t> input = {0x3B, 0x80, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00};
             const auto result = json::from_cbor(input, true, false);
-            CHECK(result.is_discarded());
+            CHECK(result.is_number_float());
         }
     }