diff --git a/src/json_stream/tests/test_tokenizer.py b/src/json_stream/tests/test_tokenizer.py
index e27b4a7..3f510d2 100644
--- a/src/json_stream/tests/test_tokenizer.py
+++ b/src/json_stream/tests/test_tokenizer.py
@@ -5,6 +5,7 @@
 
 Copyright (c) 2019 Daniel Yule
 """
+import math
 import re
 from io import StringIO
 from unittest import TestCase
@@ -59,6 +60,33 @@ def test_number_parsing(self):
         self.assertRaises(ValueError, self.tokenize_sequence, "3.6ea")
         self.assertRaises(ValueError, self.tokenize_sequence, "67.8e+a")
 
+    def test_nan_infinity_parsing(self):
+        token_list = self.tokenize_sequence("NaN")
+        self.assertEqual(1, len(token_list))
+        ttype, token = token_list[0]
+        self.assertTrue(math.isnan(token))
+        self.assertEqual(ttype, TokenType.NUMBER)
+
+        self.assertNumberEquals(float("inf"), "Infinity")
+        self.assertNumberEquals(float("-inf"), "-Infinity")
+
+        self.assertRaises(ValueError, self.tokenize_sequence, "Na")
+        self.assertRaises(ValueError, self.tokenize_sequence, "Nax")
+        self.assertRaises(ValueError, self.tokenize_sequence, "In")
+        self.assertRaises(ValueError, self.tokenize_sequence, "Inf")
+        self.assertRaises(ValueError, self.tokenize_sequence, "Infi")
+        self.assertRaises(ValueError, self.tokenize_sequence, "Infin")
+        self.assertRaises(ValueError, self.tokenize_sequence, "Infini")
+        self.assertRaises(ValueError, self.tokenize_sequence, "Infinit")
+        self.assertRaises(ValueError, self.tokenize_sequence, "Infinityx")
+        self.assertRaises(ValueError, self.tokenize_sequence, "-In")
+        self.assertRaises(ValueError, self.tokenize_sequence, "-Inf")
+        self.assertRaises(ValueError, self.tokenize_sequence, "-Infi")
+        self.assertRaises(ValueError, self.tokenize_sequence, "-Infin")
+        self.assertRaises(ValueError, self.tokenize_sequence, "-Infini")
+        self.assertRaises(ValueError, self.tokenize_sequence, "-Infinit")
+        self.assertRaises(ValueError, self.tokenize_sequence, "-Infinityx")
+
     def test_operator_parsing(self):
         self.assertOperatorEquals("{", "{")
         self.assertOperatorEquals("}", "}")
diff --git a/src/json_stream/tokenizer.py b/src/json_stream/tokenizer.py
index fbe58f8..6f501dd 100644
--- a/src/json_stream/tokenizer.py
+++ b/src/json_stream/tokenizer.py
@@ -45,6 +45,15 @@ class State:
     UNICODE_SURROGATE_START = 23
     UNICODE_SURROGATE_STRING_ESCAPE = 24
     UNICODE_SURROGATE = 25
+    NAN_1 = 26
+    NAN_2 = 27
+    INF_1 = 28
+    INF_2 = 29
+    INF_3 = 30
+    INF_4 = 31
+    INF_5 = 32
+    INF_6 = 33
+    INF_7 = 34
 
 
 class SpecialChar:
@@ -130,6 +139,10 @@ def process_char(char):
                 next_state = State.TRUE_1
             elif char == "n":
                 next_state = State.NULL_1
+            elif char == "N":
+                next_state = State.NAN_1
+            elif char == "I":
+                next_state = State.INF_1
             elif not char.isspace() and not char == SpecialChar.EOF:
                 raise ValueError("Invalid JSON character: '{0}'".format(char))
         elif state == State.INTEGER:
@@ -169,8 +182,10 @@ def process_char(char):
             elif char in "123456789":
                 next_state = State.INTEGER
                 add_char = True
+            elif char == "I":
+                next_state = State.INF_1
             else:
-                raise ValueError("A - must be followed by a digit.  Got '{0}'".format(char))
+                raise ValueError("A - must be followed by a digit or Infinity.  Got '{0}'".format(char))
         elif state == State.INTEGER_EXP_0:
             if char == "+" or char == "-" or char in "0123456789":
                 next_state = State.INTEGER_EXP
@@ -262,6 +277,55 @@ def process_char(char):
                 now_token = (TokenType.NULL, None)
             else:
                 raise ValueError("Invalid JSON character: '{0}'".format(char))
+        elif state == State.NAN_1:
+            if char == "a":
+                next_state = State.NAN_2
+            else:
+                raise ValueError("Invalid JSON character: '{0}'".format(char))
+        elif state == State.NAN_2:
+            if char == "N":
+                next_state = State.WHITESPACE
+                completed = True
+                now_token = (TokenType.NUMBER, float("NaN"))
+            else:
+                raise ValueError("Invalid JSON character: '{0}'".format(char))
+        elif state == State.INF_1:
+            if char == "n":
+                next_state = State.INF_2
+            else:
+                raise ValueError("Invalid JSON character: '{0}'".format(char))
+        elif state == State.INF_2:
+            if char == "f":
+                next_state = State.INF_3
+            else:
+                raise ValueError("Invalid JSON character: '{0}'".format(char))
+        elif state == State.INF_3:
+            if char == "i":
+                next_state = State.INF_4
+            else:
+                raise ValueError("Invalid JSON character: '{0}'".format(char))
+        elif state == State.INF_4:
+            if char == "n":
+                next_state = State.INF_5
+            else:
+                raise ValueError("Invalid JSON character: '{0}'".format(char))
+        elif state == State.INF_5:
+            if char == "i":
+                next_state = State.INF_6
+            else:
+                raise ValueError("Invalid JSON character: '{0}'".format(char))
+        elif state == State.INF_6:
+            if char == "t":
+                next_state = State.INF_7
+            else:
+                raise ValueError("Invalid JSON character: '{0}'".format(char))
+        elif state == State.INF_7:
+            if char == "y":
+                next_state = State.WHITESPACE
+                completed = True
+                now_token = (TokenType.NUMBER, float("".join(token) + "Infinity"))
+            else:
+                raise ValueError("Invalid JSON character: '{0}'".format(char))
         elif state == State.STRING:
             if char == "\"":
                 completed = True