diff --git a/doc/source/whatsnew/v0.17.0.txt b/doc/source/whatsnew/v0.17.0.txt
index e77532b2fe432..069425984b1b2 100644
--- a/doc/source/whatsnew/v0.17.0.txt
+++ b/doc/source/whatsnew/v0.17.0.txt
@@ -232,6 +232,16 @@ Other API Changes
 - Serialize metadata properties of subclasses of pandas objects (:issue:`10553`).
 
 
+- ``NaT``'s methods now either raise ``ValueError``, or return ``np.nan`` or ``NaT`` (:issue:`9513`)
+===========================     ==============================================================
+Behavior                        Methods  
+===========================     ==============================================================
+``return np.nan``               ``weekday``, ``isoweekday``
+``return NaT``                  ``date``, ``now``, ``replace``, ``to_datetime``, ``today``
+``return np.datetime64('NaT')`` ``to_datetime64`` (unchanged)   
+``raise ValueError``            All other public methods (names not beginning with underscores)
+===========================     ===============================================================
+
 .. _whatsnew_0170.deprecations:
 
 Deprecations
diff --git a/pandas/tests/test_index.py b/pandas/tests/test_index.py
index 3d901837f5123..41c814f8943b2 100644
--- a/pandas/tests/test_index.py
+++ b/pandas/tests/test_index.py
@@ -2879,6 +2879,9 @@ def test_union(self):
             result = first.union(case)
             self.assertTrue(tm.equalContents(result, everything))
 
+    def test_nat(self):
+        self.assertIs(DatetimeIndex([np.nan])[0], pd.NaT)
+
 
 class TestPeriodIndex(DatetimeLike, tm.TestCase):
     _holder = PeriodIndex
diff --git a/pandas/tests/test_series.py b/pandas/tests/test_series.py
index 7326d7a9d811d..826eddb63a5d5 100644
--- a/pandas/tests/test_series.py
+++ b/pandas/tests/test_series.py
@@ -210,7 +210,6 @@ def test_valid_dt_with_missing_values(self):
 
         # GH 8689
         s = Series(date_range('20130101',periods=5,freq='D'))
-        s_orig = s.copy()
         s.iloc[2] = pd.NaT
 
         for attr in ['microsecond','nanosecond','second','minute','hour','day']:
diff --git a/pandas/tseries/index.py b/pandas/tseries/index.py
index a8b6fb4389459..6cf972d4d7a8a 100644
--- a/pandas/tseries/index.py
+++ b/pandas/tseries/index.py
@@ -1392,7 +1392,8 @@ def time(self):
         """
         # can't call self.map() which tries to treat func as ufunc
         # and causes recursion warnings on python 2.6
-        return self._maybe_mask_results(_algos.arrmap_object(self.asobject.values, lambda x: x.time()))
+        return self._maybe_mask_results(_algos.arrmap_object(self.asobject.values,
+                                                             lambda x: np.nan if x is tslib.NaT else x.time()))
 
     @property
     def date(self):
diff --git a/pandas/tseries/tests/test_timeseries.py b/pandas/tseries/tests/test_timeseries.py
index c8b96076b26bd..a6c5dc23250b8 100644
--- a/pandas/tseries/tests/test_timeseries.py
+++ b/pandas/tseries/tests/test_timeseries.py
@@ -941,12 +941,34 @@ def test_nat_vector_field_access(self):
     def test_nat_scalar_field_access(self):
         fields = ['year', 'quarter', 'month', 'day', 'hour',
                   'minute', 'second', 'microsecond', 'nanosecond',
-                  'week', 'dayofyear', 'days_in_month']
+                  'week', 'dayofyear', 'days_in_month', 'daysinmonth',
+                  'dayofweek']
         for field in fields:
             result = getattr(NaT, field)
             self.assertTrue(np.isnan(result))
 
-        self.assertTrue(np.isnan(NaT.weekday()))
+    def test_NaT_methods(self):
+        # GH 9513
+        raise_methods = ['astimezone', 'combine', 'ctime', 'dst', 'fromordinal',
+                         'fromtimestamp', 'isocalendar', 'isoformat',
+                         'strftime', 'strptime',
+                         'time', 'timestamp', 'timetuple', 'timetz',
+                         'toordinal', 'tzname', 'utcfromtimestamp',
+                         'utcnow', 'utcoffset', 'utctimetuple']
+        nat_methods = ['date', 'now', 'replace', 'to_datetime', 'today']
+        nan_methods = ['weekday', 'isoweekday']
+
+        for method in raise_methods:
+            if hasattr(NaT, method):
+                self.assertRaises(ValueError, getattr(NaT, method))
+
+        for method in nan_methods:
+            if hasattr(NaT, method):
+                self.assertTrue(np.isnan(getattr(NaT, method)()))
+
+        for method in nat_methods:
+            if hasattr(NaT, method):
+                self.assertIs(getattr(NaT, method)(), NaT)
 
     def test_to_datetime_types(self):
 
@@ -3520,6 +3542,9 @@ def check(val,unit=None,h=1,s=1,us=0):
         result = Timestamp(NaT)
         self.assertIs(result, NaT)
 
+        result = Timestamp('NaT')
+        self.assertIs(result, NaT)
+
     def test_roundtrip(self):
 
         # test value to string and back conversions
diff --git a/pandas/tslib.pyx b/pandas/tslib.pyx
index 168bb754250e3..a2fc9b07b16a1 100644
--- a/pandas/tslib.pyx
+++ b/pandas/tslib.pyx
@@ -58,7 +58,7 @@ from dateutil.relativedelta import relativedelta
 from dateutil.parser import DEFAULTPARSER
 
 from pytz.tzinfo import BaseTzInfo as _pytz_BaseTzInfo
-from pandas.compat import parse_date, string_types, iteritems, StringIO
+from pandas.compat import parse_date, string_types, iteritems, StringIO, callable
 
 import operator
 import collections
@@ -640,15 +640,10 @@ class NaTType(_NaT):
     def __long__(self):
         return NPY_NAT
 
-    def weekday(self):
-        return np.nan
-
-    def toordinal(self):
-        return -1
-
     def __reduce__(self):
         return (__nat_unpickle, (None, ))
 
+
 fields = ['year', 'quarter', 'month', 'day', 'hour',
           'minute', 'second', 'millisecond', 'microsecond', 'nanosecond',
           'week', 'dayofyear', 'days_in_month', 'daysinmonth', 'dayofweek']
@@ -656,6 +651,50 @@ for field in fields:
     prop = property(fget=lambda self: np.nan)
     setattr(NaTType, field, prop)
 
+# GH9513 NaT methods (except to_datetime64) to raise, return np.nan, or return NaT
+# create functions that raise, for binding to NaTType
+def _make_error_func(func_name):
+    def f(*args, **kwargs):
+        raise ValueError("NaTType does not support " + func_name)
+    f.__name__ = func_name
+    return f
+
+def _make_nat_func(func_name):
+    def f(*args, **kwargs):
+        return NaT
+    f.__name__ = func_name
+    return f
+
+def _make_nan_func(func_name):
+    def f(*args, **kwargs):
+        return np.nan
+    f.__name__ = func_name
+    return f
+
+_nat_methods = ['date', 'now', 'replace', 'to_datetime', 'today']
+
+_nan_methods = ['weekday', 'isoweekday']
+
+_implemented_methods = ['to_datetime64']
+_implemented_methods.extend(_nat_methods)
+_implemented_methods.extend(_nan_methods)
+
+for _method_name in _nat_methods:
+    # not all methods exist in all versions of Python
+    if hasattr(NaTType, _method_name):
+        setattr(NaTType, _method_name, _make_nat_func(_method_name))
+
+for _method_name in _nan_methods:
+    if hasattr(NaTType, _method_name):
+        setattr(NaTType, _method_name, _make_nan_func(_method_name))
+
+for _maybe_method_name in dir(NaTType):
+    _maybe_method = getattr(NaTType, _maybe_method_name)
+    if (callable(_maybe_method)
+        and not _maybe_method_name.startswith("_")
+        and _maybe_method_name not in _implemented_methods):
+        setattr(NaTType, _maybe_method_name, _make_error_func(_maybe_method_name))
+
 def __nat_unpickle(*args):
     # return constant defined in the module
     return NaT