diff --git a/doc/source/whatsnew/v0.18.1.txt b/doc/source/whatsnew/v0.18.1.txt
index 7f74d8a769e4b..9922b326b6424 100644
--- a/doc/source/whatsnew/v0.18.1.txt
+++ b/doc/source/whatsnew/v0.18.1.txt
@@ -408,6 +408,8 @@ New behaviour:
 
    np.cumsum(sp, axis=0)
 
+- ``map`` on an ``Index`` now returns an ``Index``, not an array (:issue:`12766`)
+
 .. _whatsnew_0181.apply_resample:
 
 Using ``.apply`` on groupby resampling
diff --git a/pandas/indexes/base.py b/pandas/indexes/base.py
index 1c24a0db34b2b..70faa89a57309 100644
--- a/pandas/indexes/base.py
+++ b/pandas/indexes/base.py
@@ -2427,7 +2427,7 @@ def groupby(self, values):
 
     def map(self, mapper):
         """
-        Apply mapper function to its values.
+        Apply mapper function to an index
 
         Parameters
         ----------
@@ -2436,9 +2436,10 @@ def map(self, mapper):
 
         Returns
         -------
-        applied : array
+        An Index reflecting an appropriate Index with the mapper
+            function applied
         """
-        return self._arrmap(self.values, mapper)
+        return self._shallow_copy_with_infer(self._arrmap(self.values, mapper))
 
     def isin(self, values, level=None):
         """
diff --git a/pandas/tests/indexes/common.py b/pandas/tests/indexes/common.py
index 773f20532e4ff..98d37eb632947 100644
--- a/pandas/tests/indexes/common.py
+++ b/pandas/tests/indexes/common.py
@@ -852,3 +852,41 @@ def test_fillna(self):
                 expected[1] = True
                 self.assert_numpy_array_equal(idx._isnan, expected)
                 self.assertTrue(idx.hasnans)
+
+    def test_map(self):
+        for name, index in self.indices.items():
+            if len(index) == 0 or isinstance(index, MultiIndex):
+                pass
+            else:
+                # Applying a function to the Index
+                index = index.map(lambda x: x)
+                print(name, index)
+                self.assertTrue(isinstance(index, Index))
+                #self.assertTrue(index.equals(Index(['I1', 'I2'])))
+                #self.assertEqual(index.name, "Numbering")
+#
+                #testIdx = self.unicodeIndex.map(lambda x: len(x))
+                #tm.assert_index_equal(testIdx, Int64Index([10]*100))
+#
+                #testIdx = self.strIndex.map(lambda x: len(x))
+                #tm.assert_index_equal(testIdx, Int64Index([10]*100))
+#
+                #testIdx = self.dateIndex.map(lambda x: x + timedelta(days=1))
+                #tm.assert_index_equal(
+                #    testIdx, DatetimeIndex([dt + timedelta(days=1) for dt in tm.makeDateIndex(100)]))
+#
+                #testIdx = self.periodIndex.map(lambda x: x.to_timestamp())
+                #tm.assert_index_equal(testIdx, self.dateIndex)
+#
+                #testIdx = self.intIndex.map(lambda x: str(x))
+                #tm.assert_index_equal(testIdx, Index([str(i) for i in range(100)]))
+#
+                #testIdx = self.floatIndex.map(lambda x: -1 if x < 0 else 1)
+                #self.assertEqual(len(testIdx), 100)
+                #self.assertTrue(isinstance(testIdx, Int64Index))
+                #self.assertTrue(set(testIdx == {-1, 1}))
+#
+                #testIdx = self.boolIndex.map(lambda x: not x)
+                #tm.assert_index_equal(testIdx, Index([False, True]))
+#
+                #testIdx = self.catIndex.map(lambda x: len(x))
diff --git a/pandas/tests/indexes/test_datetimelike.py b/pandas/tests/indexes/test_datetimelike.py
index 7502a4ce26b04..34758a2b7f04b 100644
--- a/pandas/tests/indexes/test_datetimelike.py
+++ b/pandas/tests/indexes/test_datetimelike.py
@@ -1149,3 +1149,5 @@ def test_fillna_timedelta(self):
         exp = pd.Index(
             [pd.Timedelta('1 day'), 'x', pd.Timedelta('3 day')], dtype=object)
         self.assert_index_equal(idx.fillna('x'), exp)
+
+