diff --git a/doc/source/whatsnew/v0.23.0.txt b/doc/source/whatsnew/v0.23.0.txt
index e19aedac80213..3f7c4b3b0ccb7 100644
--- a/doc/source/whatsnew/v0.23.0.txt
+++ b/doc/source/whatsnew/v0.23.0.txt
@@ -827,6 +827,7 @@ Other API Changes
 - A :class:`Series` of ``dtype=category`` constructed from an empty ``dict`` will now have categories of ``dtype=object`` rather than ``dtype=float64``, consistently with the case in which an empty list is passed (:issue:`18515`)
 - All-NaN levels in a ``MultiIndex`` are now assigned ``float`` rather than ``object`` dtype, promoting consistency with ``Index`` (:issue:`17929`).
 - Levels names of a ``MultiIndex`` (when not None) are now required to be unique: trying to create a ``MultiIndex`` with repeated names will raise a ``ValueError`` (:issue:`18872`)
+- Both construction and renaming of ``Index``/``MultiIndex`` with non-hashable ``name``/``names`` will now raise ``TypeError`` (:issue:`20527`)
 - :func:`Index.map` can now accept ``Series`` and dictionary input objects (:issue:`12756`, :issue:`18482`, :issue:`18509`).
 - :func:`DataFrame.unstack` will now default to filling with ``np.nan`` for ``object`` columns. (:issue:`12815`)
 - :class:`IntervalIndex` constructor will raise if the ``closed`` parameter conflicts with how the input data is inferred to be closed (:issue:`18421`)
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 12bb09e8f8a8a..f392a716d9e5b 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -42,6 +42,7 @@
     is_datetime64_any_dtype,
     is_datetime64tz_dtype,
     is_timedelta64_dtype,
+    is_hashable,
     needs_i8_conversion,
     is_iterator, is_list_like,
     is_scalar)
@@ -1311,9 +1312,33 @@ def _get_names(self):
         return FrozenList((self.name, ))
 
     def _set_names(self, values, level=None):
+        """
+        Set new names on index. Each name has to be a hashable type.
+
+        Parameters
+        ----------
+        values : str or sequence
+            name(s) to set
+        level : int, level name, or sequence of int/level names (default None)
+            If the index is a MultiIndex (hierarchical), level(s) to set (None
+            for all levels).  Otherwise level must be None
+
+        Raises
+        ------
+        TypeError if each name is not hashable.
+        """
+        if not is_list_like(values):
+            raise ValueError('Names must be a list-like')
         if len(values) != 1:
             raise ValueError('Length of new names must be 1, got %d' %
                              len(values))
+
+        # GH 20527
+        # All items in 'name' need to be hashable:
+        for name in values:
+            if not is_hashable(name):
+                raise TypeError('{}.name must be a hashable type'
+                                .format(self.__class__.__name__))
         self.name = values[0]
 
     names = property(fset=_set_names, fget=_get_names)
@@ -1339,9 +1364,9 @@ def set_names(self, names, level=None, inplace=False):
         Examples
         --------
         >>> Index([1, 2, 3, 4]).set_names('foo')
-        Int64Index([1, 2, 3, 4], dtype='int64')
+        Int64Index([1, 2, 3, 4], dtype='int64', name='foo')
         >>> Index([1, 2, 3, 4]).set_names(['foo'])
-        Int64Index([1, 2, 3, 4], dtype='int64')
+        Int64Index([1, 2, 3, 4], dtype='int64', name='foo')
         >>> idx = MultiIndex.from_tuples([(1, u'one'), (1, u'two'),
                                           (2, u'one'), (2, u'two')],
                                           names=['foo', 'bar'])
@@ -1354,6 +1379,7 @@ def set_names(self, names, level=None, inplace=False):
                    labels=[[0, 0, 1, 1], [0, 1, 0, 1]],
                    names=[u'baz', u'bar'])
         """
+
         if level is not None and self.nlevels == 1:
             raise ValueError('Level must be None for non-MultiIndex')
 
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 8098f7bb7d246..fbcf06a28c1e5 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -16,6 +16,7 @@
     _ensure_platform_int,
     is_categorical_dtype,
     is_object_dtype,
+    is_hashable,
     is_iterator,
     is_list_like,
     pandas_dtype,
@@ -634,12 +635,29 @@ def _get_names(self):
 
     def _set_names(self, names, level=None, validate=True):
         """
+        Set new names on index. Each name has to be a hashable type.
+
+        Parameters
+        ----------
+        values : str or sequence
+            name(s) to set
+        level : int, level name, or sequence of int/level names (default None)
+            If the index is a MultiIndex (hierarchical), level(s) to set (None
+            for all levels).  Otherwise level must be None
+        validate : boolean, default True
+            validate that the names match level lengths
+
+        Raises
+        ------
+        TypeError if each name is not hashable.
+
+        Notes
+        -----
         sets names on levels. WARNING: mutates!
 
         Note that you generally want to set this *after* changing levels, so
         that it only acts on copies
         """
-
         # GH 15110
         # Don't allow a single string for names in a MultiIndex
         if names is not None and not is_list_like(names):
@@ -662,10 +680,20 @@ def _set_names(self, names, level=None, validate=True):
 
         # set the name
         for l, name in zip(level, names):
-            if name is not None and name in used:
-                raise ValueError('Duplicated level name: "{}", assigned to '
-                                 'level {}, is already used for level '
-                                 '{}.'.format(name, l, used[name]))
+            if name is not None:
+
+                # GH 20527
+                # All items in 'names' need to be hashable:
+                if not is_hashable(name):
+                    raise TypeError('{}.name must be a hashable type'
+                                    .format(self.__class__.__name__))
+
+                if name in used:
+                    raise ValueError(
+                        'Duplicated level name: "{}", assigned to '
+                        'level {}, is already used for level '
+                        '{}.'.format(name, l, used[name]))
+
             self.levels[l].rename(name, inplace=True)
             used[name] = l
 
diff --git a/pandas/tests/frame/test_indexing.py b/pandas/tests/frame/test_indexing.py
index a8b81b1b03552..8e10e4c4fbc65 100644
--- a/pandas/tests/frame/test_indexing.py
+++ b/pandas/tests/frame/test_indexing.py
@@ -125,12 +125,12 @@ def test_getitem_list(self):
         # tuples
         df = DataFrame(randn(8, 3),
                        columns=Index([('foo', 'bar'), ('baz', 'qux'),
-                                      ('peek', 'aboo')], name=['sth', 'sth2']))
+                                      ('peek', 'aboo')], name=('sth', 'sth2')))
 
         result = df[[('foo', 'bar'), ('baz', 'qux')]]
         expected = df.iloc[:, :2]
         assert_frame_equal(result, expected)
-        assert result.columns.names == ['sth', 'sth2']
+        assert result.columns.names == ('sth', 'sth2')
 
     def test_getitem_callable(self):
         # GH 12533
diff --git a/pandas/tests/indexes/test_base.py b/pandas/tests/indexes/test_base.py
index 7e19de4cca292..682517f5a6fb1 100644
--- a/pandas/tests/indexes/test_base.py
+++ b/pandas/tests/indexes/test_base.py
@@ -435,6 +435,24 @@ def test_constructor_empty(self):
         assert isinstance(empty, MultiIndex)
         assert not len(empty)
 
+    def test_constructor_nonhashable_name(self, indices):
+        # GH 20527
+
+        if isinstance(indices, MultiIndex):
+            pytest.skip("multiindex handled in test_multi.py")
+
+        name = ['0']
+        message = "Index.name must be a hashable type"
+        tm.assert_raises_regex(TypeError, message, name=name)
+
+        # With .rename()
+        renamed = [['1']]
+        tm.assert_raises_regex(TypeError, message,
+                               indices.rename, name=renamed)
+        # With .set_names()
+        tm.assert_raises_regex(TypeError, message,
+                               indices.set_names, names=renamed)
+
     def test_view_with_args(self):
 
         restricted = ['unicodeIndex', 'strIndex', 'catIndex', 'boolIndex',
diff --git a/pandas/tests/indexes/test_multi.py b/pandas/tests/indexes/test_multi.py
index 984f37042d600..88dc4cbaf7bb3 100644
--- a/pandas/tests/indexes/test_multi.py
+++ b/pandas/tests/indexes/test_multi.py
@@ -615,8 +615,27 @@ def test_constructor_mismatched_label_levels(self):
         with tm.assert_raises_regex(ValueError, label_error):
             self.index.copy().set_labels([[0, 0, 0, 0], [0, 0]])
 
-    @pytest.mark.parametrize('names', [['a', 'b', 'a'], [1, 1, 2],
-                                       [1, 'a', 1]])
+    def test_constructor_nonhashable_names(self):
+        # GH 20527
+        levels = [[1, 2], [u'one', u'two']]
+        labels = [[0, 0, 1, 1], [0, 1, 0, 1]]
+        names = ((['foo'], ['bar']))
+        message = "MultiIndex.name must be a hashable type"
+        tm.assert_raises_regex(TypeError, message,
+                               MultiIndex, levels=levels,
+                               labels=labels, names=names)
+
+        # With .rename()
+        mi = MultiIndex(levels=[[1, 2], [u'one', u'two']],
+                        labels=[[0, 0, 1, 1], [0, 1, 0, 1]],
+                        names=('foo', 'bar'))
+        renamed = [['foor'], ['barr']]
+        tm.assert_raises_regex(TypeError, message, mi.rename, names=renamed)
+        # With .set_names()
+        tm.assert_raises_regex(TypeError, message, mi.set_names, names=renamed)
+
+    @pytest.mark.parametrize('names', [['a', 'b', 'a'], ['1', '1', '2'],
+                                       ['1', 'a', '1']])
     def test_duplicate_level_names(self, names):
         # GH18872
         pytest.raises(ValueError, pd.MultiIndex.from_product,