diff --git a/py-polars/polars/dataframe/frame.py b/py-polars/polars/dataframe/frame.py
index bdf256af1b1c3..666d88e822145 100644
--- a/py-polars/polars/dataframe/frame.py
+++ b/py-polars/polars/dataframe/frame.py
@@ -5207,14 +5207,19 @@ def pipe(
 
     def with_row_index(self, name: str = "index", offset: int = 0) -> Self:
         """
-        Add a column at index 0 that counts the rows.
+        Add a row index as the first column in the DataFrame.
 
         Parameters
         ----------
         name
-            Name of the column to add.
+            Name of the index column.
         offset
-            Start the row count at this offset. Default = 0
+            Start the index at this offset. Cannot be negative.
+
+        Notes
+        -----
+        The resulting column does not have any special properties. It is a regular
+        column of type `UInt32` (or `UInt64` in `polars-u64-idx`).
 
         Examples
         --------
@@ -5235,8 +5240,24 @@ def with_row_index(self, name: str = "index", offset: int = 0) -> Self:
         │ 1     ┆ 3   ┆ 4   │
         │ 2     ┆ 5   ┆ 6   │
         └───────┴─────┴─────┘
+        >>> df.with_row_index("id", offset=1000)
+        shape: (3, 3)
+        ┌──────┬─────┬─────┐
+        │ id   ┆ a   ┆ b   │
+        │ ---  ┆ --- ┆ --- │
+        │ u32  ┆ i64 ┆ i64 │
+        ╞══════╪═════╪═════╡
+        │ 1000 ┆ 1   ┆ 2   │
+        │ 1001 ┆ 3   ┆ 4   │
+        │ 1002 ┆ 5   ┆ 6   │
+        └──────┴─────┴─────┘
         """
-        return self._from_pydf(self._df.with_row_index(name, offset))
+        try:
+            return self._from_pydf(self._df.with_row_index(name, offset))
+        except OverflowError:
+            issue = "negative" if offset < 0 else "greater than the maximum index value"
+            msg = f"`offset` input for `with_row_index` cannot be {issue}, got {offset}"
+            raise ValueError(msg) from None
 
     @deprecate_function(
         "Use `with_row_index` instead."
diff --git a/py-polars/polars/lazyframe/frame.py b/py-polars/polars/lazyframe/frame.py
index d0594b5f21f43..448ef652cd5e7 100644
--- a/py-polars/polars/lazyframe/frame.py
+++ b/py-polars/polars/lazyframe/frame.py
@@ -4565,20 +4565,25 @@ def approx_n_unique(self) -> Self:
 
     def with_row_index(self, name: str = "index", offset: int = 0) -> Self:
         """
-        Add a column at index 0 with the row number.
+        Add a row index as the first column in the LazyFrame.
 
         Parameters
         ----------
         name
-            Name of the column to add.
+            Name of the index column.
         offset
-            Start the row count at this offset.
+            Start the index at this offset. Cannot be negative.
 
         Warnings
         --------
         Using this function can have a negative effect on query performance.
         This may, for instance, block predicate pushdown optimization.
 
+        Notes
+        -----
+        The resulting column does not have any special properties. It is a regular
+        column of type `UInt32` (or `UInt64` in `polars-u64-idx`).
+
         Examples
         --------
         >>> lf = pl.LazyFrame(
@@ -4598,6 +4603,17 @@ def with_row_index(self, name: str = "index", offset: int = 0) -> Self:
         │ 1     ┆ 3   ┆ 4   │
         │ 2     ┆ 5   ┆ 6   │
         └───────┴─────┴─────┘
+        >>> lf.with_row_index("id", offset=1000).collect()
+        shape: (3, 3)
+        ┌──────┬─────┬─────┐
+        │ id   ┆ a   ┆ b   │
+        │ ---  ┆ --- ┆ --- │
+        │ u32  ┆ i64 ┆ i64 │
+        ╞══════╪═════╪═════╡
+        │ 1000 ┆ 1   ┆ 2   │
+        │ 1001 ┆ 3   ┆ 4   │
+        │ 1002 ┆ 5   ┆ 6   │
+        └──────┴─────┴─────┘
         """
         return self._from_pyldf(self._ldf.with_row_index(name, offset))