diff --git a/doc/source/whatsnew/v0.18.2.txt b/doc/source/whatsnew/v0.18.2.txt
index 80ed23b04738a..ac9c1cf109dc0 100644
--- a/doc/source/whatsnew/v0.18.2.txt
+++ b/doc/source/whatsnew/v0.18.2.txt
@@ -75,3 +75,30 @@ Performance Improvements
 
 Bug Fixes
 ~~~~~~~~~
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+
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+
+
+
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+
+
+
+- Bug in ``PeriodIndex`` and ``Period`` subtraction raises ``AttributeError`` (:issue:`13071`)
+
+
+
+
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+
+
+- Bug in ``NaT`` - ``Period`` raises ``AttributeError`` (:issue:`13071`)
diff --git a/pandas/src/period.pyx b/pandas/src/period.pyx
index e5802ccef7495..457ca2b8ec842 100644
--- a/pandas/src/period.pyx
+++ b/pandas/src/period.pyx
@@ -826,26 +826,36 @@ cdef class Period(object):
             return NotImplemented
 
     def __sub__(self, other):
-        if isinstance(other, (timedelta, np.timedelta64,
-                              offsets.Tick, offsets.DateOffset, Timedelta)):
-            neg_other = -other
-            return self + neg_other
-        elif lib.is_integer(other):
-            if self.ordinal == tslib.iNaT:
-                ordinal = self.ordinal
-            else:
-                ordinal = self.ordinal - other * self.freq.n
-            return Period(ordinal=ordinal, freq=self.freq)
+        if isinstance(self, Period):
+            if isinstance(other, (timedelta, np.timedelta64,
+                                  offsets.Tick, offsets.DateOffset, Timedelta)):
+                neg_other = -other
+                return self + neg_other
+            elif lib.is_integer(other):
+                if self.ordinal == tslib.iNaT:
+                    ordinal = self.ordinal
+                else:
+                    ordinal = self.ordinal - other * self.freq.n
+                return Period(ordinal=ordinal, freq=self.freq)
+            elif isinstance(other, Period):
+                if other.freq != self.freq:
+                    raise ValueError("Cannot do arithmetic with "
+                                     "non-conforming periods")
+                if self.ordinal == tslib.iNaT or other.ordinal == tslib.iNaT:
+                    return Period(ordinal=tslib.iNaT, freq=self.freq)
+                return self.ordinal - other.ordinal
+            elif getattr(other, '_typ', None) == 'periodindex':
+                return -other.__sub__(self)
+            else:  # pragma: no cover
+                return NotImplemented
         elif isinstance(other, Period):
-            if other.freq != self.freq:
-                raise ValueError("Cannot do arithmetic with "
-                                 "non-conforming periods")
-            if self.ordinal == tslib.iNaT or other.ordinal == tslib.iNaT:
-                return Period(ordinal=tslib.iNaT, freq=self.freq)
-            return self.ordinal - other.ordinal
-        else:  # pragma: no cover
+            if self is tslib.NaT:
+                return tslib.NaT
+            return NotImplemented
+        else:
             return NotImplemented
 
+
     def asfreq(self, freq, how='E'):
         """
         Convert Period to desired frequency, either at the start or end of the
diff --git a/pandas/tseries/base.py b/pandas/tseries/base.py
index 185d806a64fe8..f1e061bb1b9bc 100644
--- a/pandas/tseries/base.py
+++ b/pandas/tseries/base.py
@@ -14,6 +14,7 @@
                                 AbstractMethodError)
 import pandas.formats.printing as printing
 import pandas.tslib as tslib
+import pandas._period as prlib
 import pandas.lib as lib
 from pandas.core.index import Index
 from pandas.indexes.base import _index_shared_docs
@@ -533,6 +534,9 @@ def _add_datelike(self, other):
     def _sub_datelike(self, other):
         raise AbstractMethodError(self)
 
+    def _sub_period(self, other):
+        return NotImplemented
+
     @classmethod
     def _add_datetimelike_methods(cls):
         """
@@ -591,6 +595,8 @@ def __sub__(self, other):
                 return self.shift(-other)
             elif isinstance(other, (tslib.Timestamp, datetime)):
                 return self._sub_datelike(other)
+            elif isinstance(other, prlib.Period):
+                return self._sub_period(other)
             else:  # pragma: no cover
                 return NotImplemented
         cls.__sub__ = __sub__
diff --git a/pandas/tseries/period.py b/pandas/tseries/period.py
index 478b25568d471..fb91185746181 100644
--- a/pandas/tseries/period.py
+++ b/pandas/tseries/period.py
@@ -4,6 +4,7 @@
 import pandas.tseries.frequencies as frequencies
 from pandas.tseries.frequencies import get_freq_code as _gfc
 from pandas.tseries.index import DatetimeIndex, Int64Index, Index
+from pandas.tseries.tdi import TimedeltaIndex
 from pandas.tseries.base import DatelikeOps, DatetimeIndexOpsMixin
 from pandas.tseries.tools import parse_time_string
 import pandas.tseries.offsets as offsets
@@ -595,6 +596,32 @@ def _add_delta(self, other):
         ordinal_delta = self._maybe_convert_timedelta(other)
         return self.shift(ordinal_delta)
 
+    def _sub_datelike(self, other):
+        if other is tslib.NaT:
+            new_data = np.empty(len(self), dtype=np.int64)
+            new_data.fill(tslib.iNaT)
+            return TimedeltaIndex(new_data, name=self.name)
+        return NotImplemented
+
+    def _sub_period(self, other):
+        if self.freq != other.freq:
+            msg = _DIFFERENT_FREQ_INDEX.format(self.freqstr, other.freqstr)
+            raise IncompatibleFrequency(msg)
+
+        if other.ordinal == tslib.iNaT:
+            new_data = np.empty(len(self))
+            new_data.fill(np.nan)
+        else:
+            asi8 = self.asi8
+            new_data = asi8 - other.ordinal
+
+            if self.hasnans:
+                mask = asi8 == tslib.iNaT
+                new_data = new_data.astype(np.float64)
+                new_data[mask] = np.nan
+        # result must be Int64Index or Float64Index
+        return Index(new_data, name=self.name)
+
     def shift(self, n):
         """
         Specialized shift which produces an PeriodIndex
diff --git a/pandas/tseries/tests/test_period.py b/pandas/tseries/tests/test_period.py
index 12ba0b1b1bd9b..debb3e7956488 100644
--- a/pandas/tseries/tests/test_period.py
+++ b/pandas/tseries/tests/test_period.py
@@ -3418,6 +3418,16 @@ def test_add_offset_nat(self):
                 with tm.assertRaises(ValueError):
                     p + o
 
+    def test_sub_pdnat(self):
+        # GH 13071
+        p = pd.Period('2011-01', freq='M')
+        self.assertIs(p - pd.NaT, pd.NaT)
+        self.assertIs(pd.NaT - p, pd.NaT)
+
+        p = pd.Period('NaT', freq='M')
+        self.assertIs(p - pd.NaT, pd.NaT)
+        self.assertIs(pd.NaT - p, pd.NaT)
+
     def test_sub_offset(self):
         # freq is DateOffset
         for freq in ['A', '2A', '3A']:
@@ -3614,6 +3624,48 @@ def test_pi_ops_array(self):
                            '2011-01-01 12:15:00'], freq='S', name='idx')
         self.assert_index_equal(result, exp)
 
+    def test_pi_sub_period(self):
+        # GH 13071
+        idx = PeriodIndex(['2011-01', '2011-02', '2011-03',
+                           '2011-04'], freq='M', name='idx')
+
+        result = idx - pd.Period('2012-01', freq='M')
+        exp = pd.Index([-12, -11, -10, -9], name='idx')
+        tm.assert_index_equal(result, exp)
+
+        result = pd.Period('2012-01', freq='M') - idx
+        exp = pd.Index([12, 11, 10, 9], name='idx')
+        tm.assert_index_equal(result, exp)
+
+        exp = pd.Index([np.nan, np.nan, np.nan, np.nan], name='idx')
+        tm.assert_index_equal(idx - pd.Period('NaT', freq='M'), exp)
+        tm.assert_index_equal(pd.Period('NaT', freq='M') - idx, exp)
+
+    def test_pi_sub_pdnat(self):
+        # GH 13071
+        idx = PeriodIndex(['2011-01', '2011-02', 'NaT',
+                           '2011-04'], freq='M', name='idx')
+        exp = pd.TimedeltaIndex([pd.NaT] * 4, name='idx')
+        tm.assert_index_equal(pd.NaT - idx, exp)
+        tm.assert_index_equal(idx - pd.NaT, exp)
+
+    def test_pi_sub_period_nat(self):
+        # GH 13071
+        idx = PeriodIndex(['2011-01', 'NaT', '2011-03',
+                           '2011-04'], freq='M', name='idx')
+
+        result = idx - pd.Period('2012-01', freq='M')
+        exp = pd.Index([-12, np.nan, -10, -9], name='idx')
+        tm.assert_index_equal(result, exp)
+
+        result = pd.Period('2012-01', freq='M') - idx
+        exp = pd.Index([12, np.nan, 10, 9], name='idx')
+        tm.assert_index_equal(result, exp)
+
+        exp = pd.Index([np.nan, np.nan, np.nan, np.nan], name='idx')
+        tm.assert_index_equal(idx - pd.Period('NaT', freq='M'), exp)
+        tm.assert_index_equal(pd.Period('NaT', freq='M') - idx, exp)
+
 
 class TestPeriodRepresentation(tm.TestCase):
     """