diff --git a/doc/source/whatsnew/v2.1.0.rst b/doc/source/whatsnew/v2.1.0.rst
index e860d59f2e5bd..f32598e72685e 100644
--- a/doc/source/whatsnew/v2.1.0.rst
+++ b/doc/source/whatsnew/v2.1.0.rst
@@ -80,6 +80,7 @@ Other enhancements
 - :meth:`MultiIndex.sortlevel` and :meth:`Index.sortlevel` gained a new keyword ``na_position`` (:issue:`51612`)
 - :meth:`arrays.DatetimeArray.map`, :meth:`arrays.TimedeltaArray.map` and :meth:`arrays.PeriodArray.map` can now take a ``na_action`` argument (:issue:`51644`)
 - :meth:`arrays.SparseArray.map` now supports ``na_action`` (:issue:`52096`).
+- Add :meth:`diff()` and :meth:`round()` for :class:`Index` (:issue:`19708`)
 - Add dtype of categories to ``repr`` information of :class:`CategoricalDtype` (:issue:`52179`)
 - Added to the escape mode "latex-math" preserving without escaping all characters between "\(" and "\)" in formatter (:issue:`51903`)
 - Adding ``engine_kwargs`` parameter to :meth:`DataFrame.read_excel` (:issue:`52214`)
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 0a56fa4d031d6..60b4bfd65c39d 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -6758,6 +6758,59 @@ def infer_objects(self, copy: bool = True) -> Index:
             result._references.add_index_reference(result)
         return result
 
+    def diff(self, periods: int = 1):
+        """
+        Computes the difference between consecutive values in the Index object.
+
+        If periods is greater than 1, computes the difference between values that
+        are `periods` number of positions apart.
+
+        Parameters
+        ----------
+        periods : int, optional
+            The number of positions between the current and previous
+            value to compute the difference with. Default is 1.
+
+        Returns
+        -------
+        Index
+            A new Index object with the computed differences.
+
+        Examples
+        --------
+        >>> import pandas as pd
+        >>> idx = pd.Index([10, 20, 30, 40, 50])
+        >>> idx.diff()
+        Index([nan, 10.0, 10.0, 10.0, 10.0], dtype='float64')
+
+        """
+        return self._constructor(self.to_series().diff(periods))
+
+    def round(self, decimals: int = 0):
+        """
+        Round each value in the Index to the given number of decimals.
+
+        Parameters
+        ----------
+        decimals : int, optional
+            Number of decimal places to round to. If decimals is negative,
+            it specifies the number of positions to the left of the decimal point.
+
+        Returns
+        -------
+        Index
+            A new Index with the rounded values.
+
+        Examples
+        --------
+        >>> import pandas as pd
+        >>> idx = pd.Index([10.1234, 20.5678, 30.9123, 40.4567, 50.7890])
+        >>> idx.round(decimals=2)
+        Index([10.12, 20.57, 30.91, 40.46, 50.79], dtype='float64')
+
+        """
+        return self._constructor(self.to_series().round(decimals))
+
     # --------------------------------------------------------------------
     # Generated Arithmetic, Comparison, and Unary Methods
 
diff --git a/pandas/tests/indexes/test_base.py b/pandas/tests/indexes/test_base.py
index e681223933abb..db317a819c520 100644
--- a/pandas/tests/indexes/test_base.py
+++ b/pandas/tests/indexes/test_base.py
@@ -1270,6 +1270,38 @@ def test_sortlevel_na_position(self):
         expected = Index([np.nan, 1])
         tm.assert_index_equal(result, expected)
 
+    @pytest.mark.parametrize(
+        "periods, expected_results",
+        [
+            (1, [np.nan, 10, 10, 10, 10]),
+            (2, [np.nan, np.nan, 20, 20, 20]),
+            (3, [np.nan, np.nan, np.nan, 30, 30]),
+        ],
+    )
+    def test_index_diff(self, periods, expected_results):
+        # GH#19708
+        idx = Index([10, 20, 30, 40, 50])
+        result = idx.diff(periods)
+        expected = Index(expected_results)
+
+        tm.assert_index_equal(result, expected)
+
+    @pytest.mark.parametrize(
+        "decimals, expected_results",
+        [
+            (0, [1.0, 2.0, 3.0]),
+            (1, [1.2, 2.3, 3.5]),
+            (2, [1.23, 2.35, 3.46]),
+        ],
+    )
+    def test_index_round(self, decimals, expected_results):
+        # GH#19708
+        idx = Index([1.234, 2.345, 3.456])
+        result = idx.round(decimals)
+        expected = Index(expected_results)
+
+        tm.assert_index_equal(result, expected)
+
 
 class TestMixedIntIndex(Base):
     # Mostly the tests from common.py for which the results differ