diff --git a/source/util/hex_float.h b/source/util/hex_float.h
index 903b628..06e3c57 100644
--- a/source/util/hex_float.h
+++ b/source/util/hex_float.h
@@ -209,9 +209,10 @@
 // be the default for any non-specialized type.
 template <typename T>
 struct HexFloatTraits {
-  // Integer type that can store this hex-float.
+  // Integer type that can store the bit representation of this hex-float.
   using uint_type = void;
-  // Signed integer type that can store this hex-float.
+  // Signed integer type that can store the bit representation of this
+  // hex-float.
   using int_type = void;
   // The numerical type that this HexFloat represents.
   using underlying_type = void;
@@ -958,9 +959,15 @@
   // This "looks" like a hex-float so treat it as one.
   bool seen_p = false;
   bool seen_dot = false;
+
+  // The mantissa bits, without the most significant 1 bit, and with the
+  // the most recently read bits in the least significant positions.
+  uint_type fraction = 0;
+  // The number of mantissa bits that have been read, including the leading 1
+  // bit that is not written into 'fraction'.
   uint_type fraction_index = 0;
 
-  uint_type fraction = 0;
+  // TODO(dneto): handle overflow and underflow
   int_type exponent = HF::exponent_bias;
 
   // Strip off leading zeros so we don't have to special-case them later.
@@ -968,11 +975,13 @@
     is.get();
   }
 
-  bool is_denorm =
-      true;  // Assume denorm "representation" until we hear otherwise.
-             // NB: This does not mean the value is actually denorm,
-             // it just means that it was written 0.
+  // Does the mantissa, as written, have non-zero digits to the left of
+  // the decimal point.  Assume no until proven otherwise.
+  bool has_integer_part = false;
   bool bits_written = false;  // Stays false until we write a bit.
+
+  // Scan the mantissa hex digits until we see a '.' or the 'p' that
+  // starts the exponent.
   while (!seen_p && !seen_dot) {
     // Handle characters that are left of the fractional part.
     if (next_char == '.') {
@@ -980,9 +989,8 @@
     } else if (next_char == 'p') {
       seen_p = true;
     } else if (::isxdigit(next_char)) {
-      // We know this is not denormalized since we have stripped all leading
-      // zeroes and we are not a ".".
-      is_denorm = false;
+      // We have stripped all leading zeroes and we have not yet seen a ".".
+      has_integer_part = true;
       int number = get_nibble_from_character(next_char);
       for (int i = 0; i < 4; ++i, number <<= 1) {
         uint_type write_bit = (number & 0x8) ? 0x1 : 0x0;
@@ -993,8 +1001,12 @@
               fraction |
               static_cast<uint_type>(
                   write_bit << (HF::top_bit_left_shift - fraction_index++)));
+          // TODO(dneto): Avoid overflow. Testing would require
+          // parameterization.
           exponent = static_cast<int_type>(exponent + 1);
         }
+        // Since this updated after setting fraction bits, this effectively
+        // drops the leading 1 bit.
         bits_written |= write_bit != 0;
       }
     } else {
@@ -1018,10 +1030,12 @@
       for (int i = 0; i < 4; ++i, number <<= 1) {
         uint_type write_bit = (number & 0x8) ? 0x01 : 0x00;
         bits_written |= write_bit != 0;
-        if (is_denorm && !bits_written) {
+        if ((!has_integer_part) && !bits_written) {
           // Handle modifying the exponent here this way we can handle
           // an arbitrary number of hex values without overflowing our
           // integer.
+          // TODO(dneto): Handle underflow. Testing would require extra
+          // parameterization.
           exponent = static_cast<int_type>(exponent - 1);
         } else {
           fraction = static_cast<uint_type>(
@@ -1043,25 +1057,40 @@
   // Finished reading the part preceding 'p'.
   // In hex floats syntax, the binary exponent is required.
 
-  bool seen_sign = false;
+  bool seen_exponent_sign = false;
   int8_t exponent_sign = 1;
   bool seen_written_exponent_digits = false;
+  // The magnitude of the exponent, as written, or the sentinel value to signal
+  // overflow.
   int_type written_exponent = 0;
+  // A sentinel value signalling overflow of the magnitude of the written
+  // exponent.  We'll assume that -written_exponent_overflow is valid for the
+  // type. Later we may add 1 or subtract 1 from the adjusted exponent, so leave
+  // room for an extra 1.
+  const int_type written_exponent_overflow =
+      std::numeric_limits<int_type>::max() - 1;
   while (true) {
     if (!seen_written_exponent_digits &&
         (next_char == '-' || next_char == '+')) {
-      if (seen_sign) {
+      if (seen_exponent_sign) {
         is.setstate(std::ios::failbit);
         return is;
       }
-      seen_sign = true;
+      seen_exponent_sign = true;
       exponent_sign = (next_char == '-') ? -1 : 1;
     } else if (::isdigit(next_char)) {
       seen_written_exponent_digits = true;
       // Hex-floats express their exponent as decimal.
-      written_exponent = static_cast<int_type>(written_exponent * 10);
-      written_exponent =
-          static_cast<int_type>(written_exponent + (next_char - '0'));
+      int_type digit =
+          static_cast<int_type>(static_cast<int_type>(next_char) - '0');
+      if (written_exponent >= (written_exponent_overflow - digit) / 10) {
+        // The exponent is very big. Saturate rather than overflow the exponent.
+        // signed integer, which would be undefined behaviour.
+        written_exponent = written_exponent_overflow;
+      } else {
+        written_exponent = static_cast<int_type>(
+            static_cast<int_type>(written_exponent * 10) + digit);
+      }
     } else {
       break;
     }
@@ -1075,10 +1104,29 @@
   }
 
   written_exponent = static_cast<int_type>(written_exponent * exponent_sign);
-  exponent = static_cast<int_type>(exponent + written_exponent);
+  // Now fold in the exponent bias into the written exponent, updating exponent.
+  // But avoid undefined behaviour that would result from overflowing int_type.
+  if (written_exponent >= 0 && exponent >= 0) {
+    // Saturate up to written_exponent_overflow.
+    if (written_exponent_overflow - exponent > written_exponent) {
+      exponent = static_cast<int_type>(written_exponent + exponent);
+    } else {
+      exponent = written_exponent_overflow;
+    }
+  } else if (written_exponent < 0 && exponent < 0) {
+    // Saturate down to -written_exponent_overflow.
+    if (written_exponent_overflow + exponent > -written_exponent) {
+      exponent = static_cast<int_type>(written_exponent + exponent);
+    } else {
+      exponent = static_cast<int_type>(-written_exponent_overflow);
+    }
+  } else {
+    // They're of opposing sign, so it's safe to add.
+    exponent = static_cast<int_type>(written_exponent + exponent);
+  }
 
-  bool is_zero = is_denorm && (fraction == 0);
-  if (is_denorm && !is_zero) {
+  bool is_zero = (!has_integer_part) && (fraction == 0);
+  if ((!has_integer_part) && !is_zero) {
     fraction = static_cast<uint_type>(fraction << 1);
     exponent = static_cast<int_type>(exponent - 1);
   } else if (is_zero) {
@@ -1095,7 +1143,7 @@
   const int_type max_exponent =
       SetBits<uint_type, 0, HF::num_exponent_bits>::get;
 
-  // Handle actual denorm numbers
+  // Handle denorm numbers
   while (exponent < 0 && !is_zero) {
     fraction = static_cast<uint_type>(fraction >> 1);
     exponent = static_cast<int_type>(exponent + 1);
diff --git a/test/hex_float_test.cpp b/test/hex_float_test.cpp
index 7edfd43..25d3c70 100644
--- a/test/hex_float_test.cpp
+++ b/test/hex_float_test.cpp
@@ -1395,6 +1395,47 @@
         {"0x1.0p1+", true, "+", 2.0f},
         {"0x1.0p1-", true, "-", 2.0f}}));
 
+INSTANTIATE_TEST_SUITE_P(
+    HexFloatPositiveExponentOverflow, FloatStreamParseTest,
+    ::testing::ValuesIn(std::vector<StreamParseCase<float>>{
+        // Positive exponents
+        {"0x1.0p1", true, "", 2.0f},       // fine, a normal number
+        {"0x1.0p15", true, "", 32768.0f},  // fine, a normal number
+        {"0x1.0p127", true, "", float(ldexp(1.0f, 127))},   // good large number
+        {"0x0.8p128", true, "", float(ldexp(1.0f, 127))},   // good large number
+        {"0x0.1p131", true, "", float(ldexp(1.0f, 127))},   // good large number
+        {"0x0.01p135", true, "", float(ldexp(1.0f, 127))},  // good large number
+        {"0x1.0p128", true, "", float(ldexp(1.0f, 128))},   // infinity
+        {"0x1.0p4294967295", true, "", float(ldexp(1.0f, 128))},  // infinity
+        {"0x1.0p5000000000", true, "", float(ldexp(1.0f, 128))},  // infinity
+        {"0x0.0p5000000000", true, "", 0.0f},  // zero mantissa, zero result
+    }));
+
+INSTANTIATE_TEST_SUITE_P(
+    HexFloatNegativeExponentOverflow, FloatStreamParseTest,
+    ::testing::ValuesIn(std::vector<StreamParseCase<float>>{
+        // Positive results, digits before '.'
+        {"0x1.0p-126", true, "",
+         float(ldexp(1.0f, -126))},  // fine, a small normal number
+        {"0x1.0p-127", true, "", float(ldexp(1.0f, -127))},  // denorm number
+        {"0x1.0p-149", true, "",
+         float(ldexp(1.0f, -149))},  // smallest positive denormal
+        {"0x0.8p-148", true, "",
+         float(ldexp(1.0f, -149))},  // smallest positive denormal
+        {"0x0.1p-145", true, "",
+         float(ldexp(1.0f, -149))},  // smallest positive denormal
+        {"0x0.01p-141", true, "",
+         float(ldexp(1.0f, -149))},  // smallest positive denormal
+
+        // underflow rounds down to zero
+        {"0x1.0p-150", true, "", 0.0f},
+        {"0x1.0p-4294967296", true, "",
+         0.0f},  // avoid exponent overflow in parser
+        {"0x1.0p-5000000000", true, "",
+         0.0f},  // avoid exponent overflow in parser
+        {"0x0.0p-5000000000", true, "", 0.0f},  // zero mantissa, zero result
+    }));
+
 // TODO(awoloszyn): Add fp16 tests and HexFloatTraits.
 }  // namespace
 }  // namespace utils