diff --git a/xarray/core/coordinates.py b/xarray/core/coordinates.py
index 820937dae6a..e5a697f171f 100644
--- a/xarray/core/coordinates.py
+++ b/xarray/core/coordinates.py
@@ -10,7 +10,7 @@
     expand_and_merge_variables, merge_coords, merge_coords_for_inplace_math)
 from .pycompat import OrderedDict
 from .utils import Frozen, ReprObject, either_dict_or_kwargs
-from .variable import Variable
+from .variable import Variable, IndexVariable
 
 # Used as the key corresponding to a DataArray's variable when converting
 # arbitrary DataArray objects to datasets
diff --git a/xarray/core/formatting.py b/xarray/core/formatting.py
index 50fa64c9987..f68076c0732 100644
--- a/xarray/core/formatting.py
+++ b/xarray/core/formatting.py
@@ -272,7 +272,7 @@ def summarize_datavar(name, var, col_width):
 
 
 def summarize_coord(name, var, col_width):
-    is_index = name in var.dims
+    is_index = name in var.dims and var.ndim==1
     show_values = var._in_memory
     marker = u'*' if is_index else u' '
     if is_index:
diff --git a/xarray/core/indexes.py b/xarray/core/indexes.py
index ffa483fc370..8ac37f03e4d 100644
--- a/xarray/core/indexes.py
+++ b/xarray/core/indexes.py
@@ -52,4 +52,4 @@ def default_indexes(coords, dims):
     to indexes used for indexing along that dimension.
     """
     return OrderedDict((key, coords[key].to_index())
-                       for key in dims if key in coords)
+                       for key in dims if key in coords and coords[key].ndim==1)
diff --git a/xarray/core/variable.py b/xarray/core/variable.py
index 8bd7225efc3..7b75d242985 100644
--- a/xarray/core/variable.py
+++ b/xarray/core/variable.py
@@ -105,13 +105,11 @@ def as_variable(obj, name=None):
 
     if name is not None and name in obj.dims:
         # convert the Variable into an Index
-        if obj.ndim != 1:
-            raise MissingDimensionsError(
-                '%r has more than 1-dimension and the same name as one of its '
-                'dimensions %r. xarray disallows such variables because they '
-                'conflict with the coordinates used to label '
-                'dimensions.' % (name, obj.dims))
-        obj = obj.to_index_variable()
+        if obj.ndim == 1:
+            obj = obj.to_index_variable()
+        else:
+            pass
+            # TODO: do something else with multidimensional variables
 
     return obj
 
diff --git a/xarray/tests/test_dataset.py b/xarray/tests/test_dataset.py
index e55caf1bf13..5bc9489c553 100644
--- a/xarray/tests/test_dataset.py
+++ b/xarray/tests/test_dataset.py
@@ -60,6 +60,11 @@ def create_test_multiindex():
     return Dataset({}, {'x': mindex})
 
 
+def create_test_multidim_coord_no_index():
+    ds = xr.Dataset({'a': ('x', [1, 1]),
+                     'x': (['x', 'y'], [[1, 1, 1, 1], [2, 2, 2, 2]])})
+    return ds
+
 class InaccessibleVariableDataStore(backends.InMemoryDataStore):
     def __init__(self):
         super(InaccessibleVariableDataStore, self).__init__()
@@ -176,6 +181,11 @@ def test_repr_period_index(self):
         # check that creating the repr doesn't raise an error #GH645
         repr(data)
 
+    def test_repr_no_index_on_multidim_coord(self):
+        data = create_test_multidim_coord_no_index()
+        actual = repr(data)
+        assert '*' not in actual
+
     def test_unicode_data(self):
         # regression test for GH834
         data = Dataset({u'foø': [u'ba®']}, attrs={u'å': u'∑'})
@@ -237,8 +247,9 @@ def test_constructor(self):
 
         with raises_regex(ValueError, 'conflicting sizes'):
             Dataset({'a': x1, 'b': x2})
-        with raises_regex(ValueError, "disallows such variables"):
-            Dataset({'a': x1, 'x': z})
+        # these are now allowed -- let's add a more explicit test
+        #with raises_regex(ValueError, "disallows such variables"):
+        #    Dataset({'a': x1, 'x': z})
         with raises_regex(TypeError, 'tuple of form'):
             Dataset({'x': (1, 2, 3, 4, 5, 6, 7)})
         with raises_regex(ValueError, 'already exists as a scalar'):
@@ -249,6 +260,17 @@ def test_constructor(self):
         actual = Dataset({'z': expected['z']})
         assert_identical(expected, actual)
 
+    def test_constructor_multidim_dimensions(self):
+        # checks related to GH2368, GH2233
+        ds = create_test_multidim_coord_no_index()
+
+        # dataset should have no indices
+        assert not isinstance(ds.variables['x'], IndexVariable)
+        assert len(ds.indexes) == 0
+        assert 'x' not in ds.indexes
+        with pytest.raises(KeyError):
+            ds.indexes['x']
+
     def test_constructor_invalid_dims(self):
         # regression for GH1120
         with pytest.raises(MergeError):