diff --git a/geopandas/geodataframe.py b/geopandas/geodataframe.py
index 1bcc6ce..fd110f6 100644
--- a/geopandas/geodataframe.py
+++ b/geopandas/geodataframe.py
@@ -1652,6 +1652,34 @@ box': (2.0, 1.0, 2.0, 1.0)}], 'bbox': (1.0, 1.0, 2.0, 2.0)}
         # do not return a GeoDataFrame
         return pd.DataFrame(df)
 
+    def convert_dtypes(self, *args, **kwargs):
+        """
+        Convert columns to best possible dtypes using dtypes supporting ``pd.NA``.
+
+        Always returns a GeoDataFrame as no conversions are applied to the
+        geometry column.
+
+        See the pandas.DataFrame.convert_dtypes docstring for more details.
+
+        Returns
+        -------
+        GeoDataFrame
+
+        """
+        # Overridden to fix GH1870, that return type is not preserved always
+        # (and where it was, geometry col was not)
+
+        if not compat.PANDAS_GE_10:
+            raise NotImplementedError(
+                "GeoDataFrame.convert_dtypes requires pandas >= 1.0"
+            )
+
+        return GeoDataFrame(
+            super().convert_dtypes(*args, **kwargs),
+            geometry=self.geometry.name,
+            crs=self.crs,
+        )
+
     def to_postgis(
         self,
         name,
diff --git a/geopandas/tests/test_pandas_methods.py b/geopandas/tests/test_pandas_methods.py
index 3267313..2db561d 100644
--- a/geopandas/tests/test_pandas_methods.py
+++ b/geopandas/tests/test_pandas_methods.py
@@ -262,6 +262,42 @@ def test_astype_invalid_geodataframe():
     assert res["a"].dtype == object
 
 
+@pytest.mark.xfail(
+    not compat.PANDAS_GE_10,
+    reason="Convert dtypes new in pandas 1.0",
+    raises=NotImplementedError,
+)
+def test_convert_dtypes(df):
+    # https://github.com/geopandas/geopandas/issues/1870
+
+    # Test geometry col is first col, first, geom_col_name=geometry
+    # (order is important in concat, used internally)
+    res1 = df.convert_dtypes()  # note res1 done first for pandas < 1 xfail check
+
+    expected1 = GeoDataFrame(
+        pd.DataFrame(df).convert_dtypes(), crs=df.crs, geometry=df.geometry.name
+    )
+
+    # Checking type and metadata are right
+    assert_geodataframe_equal(expected1, res1)
+
+    # Test geom last, geom_col_name=geometry
+    res2 = df[["value1", "value2", "geometry"]].convert_dtypes()
+    assert_geodataframe_equal(expected1[["value1", "value2", "geometry"]], res2)
+
+    # Test again with crs set and custom geom col name
+    df2 = df.set_crs(epsg=4326).rename_geometry("points")
+    expected2 = GeoDataFrame(
+        pd.DataFrame(df2).convert_dtypes(), crs=df2.crs, geometry=df2.geometry.name
+    )
+    res3 = df2.convert_dtypes()
+    assert_geodataframe_equal(expected2, res3)
+
+    # Test geom last, geom_col=geometry
+    res4 = df2[["value1", "value2", "points"]].convert_dtypes()
+    assert_geodataframe_equal(expected2[["value1", "value2", "points"]], res4)
+
+
 def test_to_csv(df):
 
     exp = (