diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index a04ba157ce0ae..8cbc95f0349cf 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -156,7 +156,7 @@ Reshaping
 - Bug in :meth:`DataFrame.pivot_table` when ``margin`` is ``True`` and only ``column`` is defined (:issue:`31016`)
 - Fix incorrect error message in :meth:`DataFrame.pivot` when ``columns`` is set to ``None``. (:issue:`30924`)
 - Bug in :func:`crosstab` when inputs are two Series and have tuple names, the output will keep dummy MultiIndex as columns. (:issue:`18321`)
-
+- Bug in :func:`concat` where the resulting indices are not copied when ``copy=True`` (:issue:`29879`)
 
 Sparse
 ^^^^^^
diff --git a/pandas/core/indexes/api.py b/pandas/core/indexes/api.py
index 4072d06b9427c..0a23d38ace37e 100644
--- a/pandas/core/indexes/api.py
+++ b/pandas/core/indexes/api.py
@@ -63,7 +63,7 @@
 
 
 def get_objs_combined_axis(
-    objs, intersect: bool = False, axis=0, sort: bool = True
+    objs, intersect: bool = False, axis=0, sort: bool = True, copy: bool = False
 ) -> Index:
     """
     Extract combined index: return intersection or union (depending on the
@@ -81,13 +81,15 @@ def get_objs_combined_axis(
         The axis to extract indexes from.
     sort : bool, default True
         Whether the result index should come out sorted or not.
+    copy : bool, default False
+        If True, return a copy of the combined index.
 
     Returns
     -------
     Index
     """
     obs_idxes = [obj._get_axis(axis) for obj in objs]
-    return _get_combined_index(obs_idxes, intersect=intersect, sort=sort)
+    return _get_combined_index(obs_idxes, intersect=intersect, sort=sort, copy=copy)
 
 
 def _get_distinct_objs(objs: List[Index]) -> List[Index]:
@@ -105,7 +107,10 @@ def _get_distinct_objs(objs: List[Index]) -> List[Index]:
 
 
 def _get_combined_index(
-    indexes: List[Index], intersect: bool = False, sort: bool = False
+    indexes: List[Index],
+    intersect: bool = False,
+    sort: bool = False,
+    copy: bool = False,
 ) -> Index:
     """
     Return the union or intersection of indexes.
@@ -119,6 +124,8 @@ def _get_combined_index(
         calculate the union.
     sort : bool, default False
         Whether the result index should come out sorted or not.
+    copy : bool, default False
+        If True, return a copy of the combined index.
 
     Returns
     -------
@@ -143,6 +150,11 @@ def _get_combined_index(
             index = index.sort_values()
         except TypeError:
             pass
+
+    # GH 29879
+    if copy:
+        index = index.copy()
+
     return index
 
 
diff --git a/pandas/core/reshape/concat.py b/pandas/core/reshape/concat.py
index 9528de36a3664..b42497b507e1f 100644
--- a/pandas/core/reshape/concat.py
+++ b/pandas/core/reshape/concat.py
@@ -517,7 +517,11 @@ def _get_new_axes(self) -> List[Index]:
     def _get_comb_axis(self, i: int) -> Index:
         data_axis = self.objs[0]._get_block_manager_axis(i)
         return get_objs_combined_axis(
-            self.objs, axis=data_axis, intersect=self.intersect, sort=self.sort
+            self.objs,
+            axis=data_axis,
+            intersect=self.intersect,
+            sort=self.sort,
+            copy=self.copy,
         )
 
     def _get_concat_axis(self) -> Index:
diff --git a/pandas/tests/reshape/test_concat.py b/pandas/tests/reshape/test_concat.py
index b3b2c5a05c6ad..5811f3bc196a1 100644
--- a/pandas/tests/reshape/test_concat.py
+++ b/pandas/tests/reshape/test_concat.py
@@ -2750,3 +2750,17 @@ def test_concat_sparse():
     )
     result = pd.concat([a, a], axis=1)
     tm.assert_frame_equal(result, expected)
+
+
+@pytest.mark.parametrize("test_series", [True, False])
+def test_concat_copy_index(test_series, axis):
+    # GH 29879
+    if test_series:
+        ser = Series([1, 2])
+        comb = concat([ser, ser], axis=axis, copy=True)
+        assert comb.index is not ser.index
+    else:
+        df = DataFrame([[1, 2], [3, 4]], columns=["a", "b"])
+        comb = concat([df, df], axis=axis, copy=True)
+        assert comb.index is not df.index
+        assert comb.columns is not df.columns