diff --git a/narwhals/__init__.py b/narwhals/__init__.py
index a910f0e0e4..6854b86763 100644
--- a/narwhals/__init__.py
+++ b/narwhals/__init__.py
@@ -13,6 +13,11 @@
     maybe_reset_index,
     maybe_set_index,
 )
+from narwhals.api import (
+    register_dataframe_namespace,
+    register_expr_namespace,
+    register_lazyframe_namespace,
+)
 from narwhals.dataframe import DataFrame, LazyFrame
 from narwhals.dtypes import (
     Array,
@@ -161,6 +166,9 @@
     "nth",
     "read_csv",
     "read_parquet",
+    "register_dataframe_namespace",
+    "register_expr_namespace",
+    "register_lazyframe_namespace",
     "scan_csv",
     "scan_parquet",
     "selectors",
diff --git a/narwhals/_reexport.py b/narwhals/_reexport.py
new file mode 100644
index 0000000000..42d7fe471e
--- /dev/null
+++ b/narwhals/_reexport.py
@@ -0,0 +1,8 @@
+"""Re-export Narwhals functionality to avoid cyclical imports."""
+from __future__ import annotations
+
+from narwhals.dataframe import DataFrame, LazyFrame
+from narwhals.expr import Expr
+from narwhals.series import Series
+
+__all__ = ["DataFrame", "Expr", "LazyFrame", "Series"]
diff --git a/narwhals/api.py b/narwhals/api.py
new file mode 100644
index 0000000000..fc1bf6fec1
--- /dev/null
+++ b/narwhals/api.py
@@ -0,0 +1,366 @@
+from __future__ import annotations
+
+from typing import TYPE_CHECKING, Callable, Generic, TypeVar
+from warnings import warn
+
+import narwhals._reexport as nw
+from narwhals._utils import find_stacklevel
+
+if TYPE_CHECKING:
+    from narwhals import DataFrame, Expr, LazyFrame, Series
+
+
+__all__ = [
+    "register_dataframe_namespace",
+    "register_expr_namespace",
+    "register_lazyframe_namespace",
+    "register_series_namespace",
+]
+
+# do not allow override of Narwhals' own namespaces (as registered by '_accessors')
+_reserved_namespaces: set[str] = set.union(
+    *(cls._accessors for cls in (nw.DataFrame, nw.Expr, nw.LazyFrame, nw.Series))
+)
+
+
+NS = TypeVar("NS")
+
+
+class NameSpace(Generic[NS]):
+    """Establish property-like namespace object for user-defined functionality."""
+
+    def __init__(self, name: str, namespace: type[NS]) -> None:
+        self._accessor = name
+        self._ns = namespace
+
+    def __get__(self, instance: NS | None, cls: type[NS]) -> NS | type[NS]:
+        if instance is None:
+            return self._ns
+
+        ns_instance = self._ns(instance)  # type: ignore[call-arg]
+        setattr(instance, self._accessor, ns_instance)
+        return ns_instance
+
+
+def _create_namespace(
+    name: str, cls: type[Expr | DataFrame | LazyFrame | Series]
+) -> Callable[[type[NS]], type[NS]]:
+    """Register custom namespace against the underlying Narwhals class."""
+
+    def namespace(ns_class: type[NS]) -> type[NS]:
+        if name in _reserved_namespaces:
+            msg = f"cannot override reserved namespace {name!r}"
+            raise AttributeError(msg)
+        if hasattr(cls, name):
+            warn(
+                f"Overriding existing custom namespace {name!r} (on {cls.__name__!r})",
+                UserWarning,
+                stacklevel=find_stacklevel(),
+            )
+
+        setattr(cls, name, NameSpace(name, ns_class))
+        cls._accessors.add(name)
+        return ns_class
+
+    return namespace
+
+
+def register_expr_namespace(name: str) -> Callable[[type[NS]], type[NS]]:
+    """Decorator for registering custom functionality with a Narwhals Expr.
+
+    Parameters
+    ----------
+    name
+        Name under which the functionality will be accessed.
+
+    See Also:
+    --------
+    register_dataframe_namespace : Register functionality on a DataFrame.
+    register_lazyframe_namespace : Register functionality on a LazyFrame.
+    register_series_namespace : Register functionality on a Series.
+
+    Examples:
+    --------
+    >>> @nw.api.register_expr_namespace("pow_n")
+    ... class PowersOfN:
+    ...     def __init__(self, expr: nw.Expr) -> None:
+    ...         self._expr = expr
+    ...
+    ...     def next(self, p: int) -> nw.Expr:
+    ...         return (p ** (self._expr.log(p).ceil()).cast(nw.Int64)).cast(nw.Int64)
+    ...
+    ...     def previous(self, p: int) -> nw.Expr:
+    ...         return (p ** (self._expr.log(p).floor()).cast(nw.Int64)).cast(nw.Int64)
+    ...
+    ...     def nearest(self, p: int) -> nw.Expr:
+    ...         return (p ** (self._expr.log(p)).round(0).cast(nw.Int64)).cast(nw.Int64)
+    >>>
+    >>> df = nw.DataFrame([1.4, 24.3, 55.0, 64.001], schema=["n"])
+    >>> df.select(
+    ...     nw.col("n"),
+    ...     nw.col("n").pow_n.next(p=2).alias("next_pow2"),
+    ...     nw.col("n").pow_n.previous(p=2).alias("prev_pow2"),
+    ...     nw.col("n").pow_n.nearest(p=2).alias("nearest_pow2"),
+    ... )
+    shape: (4, 4)
+    ┌────────┬───────────┬───────────┬──────────────┐
+    │ n      ┆ next_pow2 ┆ prev_pow2 ┆ nearest_pow2 │
+    │ ---    ┆ ---       ┆ ---       ┆ ---          │
+    │ f64    ┆ i64       ┆ i64       ┆ i64          │
+    ╞════════╪═══════════╪═══════════╪══════════════╡
+    │ 1.4    ┆ 2         ┆ 1         ┆ 1            │
+    │ 24.3   ┆ 32        ┆ 16        ┆ 32           │
+    │ 55.0   ┆ 64        ┆ 32        ┆ 64           │
+    │ 64.001 ┆ 128       ┆ 64        ┆ 64           │
+    └────────┴───────────┴───────────┴──────────────┘
+    """
+    return _create_namespace(name, nw.Expr)
+
+
+def register_dataframe_namespace(name: str) -> Callable[[type[NS]], type[NS]]:
+    """Decorator for registering custom functionality with a Narwhals DataFrame.
+
+    Parameters
+    ----------
+    name
+        Name under which the functionality will be accessed.
+
+    See Also:
+    --------
+    register_expr_namespace : Register functionality on an Expr.
+    register_lazyframe_namespace : Register functionality on a LazyFrame.
+    register_series_namespace : Register functionality on a Series.
+
+    Examples:
+    --------
+    >>> @nw.api.register_dataframe_namespace("split")
+    ... class SplitFrame:
+    ...     def __init__(self, df: nw.DataFrame) -> None:
+    ...         self._df = df
+    ...
+    ...     def by_first_letter_of_column_names(self) -> list[nw.DataFrame]:
+    ...         return [
+    ...             self._df.select([col for col in self._df.columns if col[0] == f])
+    ...             for f in dict.fromkeys(col[0] for col in self._df.columns)
+    ...         ]
+    ...
+    ...     def by_first_letter_of_column_values(self, col: str) -> list[nw.DataFrame]:
+    ...         return [
+    ...             self._df.filter(nw.col(col).str.starts_with(c))
+    ...             for c in sorted(
+    ...                 set(df.select(nw.col(col).str.slice(0, 1)).to_series())
+    ...             )
+    ...         ]
+    >>>
+    >>> df = nw.DataFrame(
+    ...     data=[["xx", 2, 3, 4], ["xy", 4, 5, 6], ["yy", 5, 6, 7], ["yz", 6, 7, 8]],
+    ...     schema=["a1", "a2", "b1", "b2"],
+    ...     orient="row",
+    ... )
+    >>> df
+    shape: (4, 4)
+    ┌─────┬─────┬─────┬─────┐
+    │ a1  ┆ a2  ┆ b1  ┆ b2  │
+    │ --- ┆ --- ┆ --- ┆ --- │
+    │ str ┆ i64 ┆ i64 ┆ i64 │
+    ╞═════╪═════╪═════╪═════╡
+    │ xx  ┆ 2   ┆ 3   ┆ 4   │
+    │ xy  ┆ 4   ┆ 5   ┆ 6   │
+    │ yy  ┆ 5   ┆ 6   ┆ 7   │
+    │ yz  ┆ 6   ┆ 7   ┆ 8   │
+    └─────┴─────┴─────┴─────┘
+    >>> df.split.by_first_letter_of_column_names()
+    [shape: (4, 2)
+    ┌─────┬─────┐
+    │ a1  ┆ a2  │
+    │ --- ┆ --- │
+    │ str ┆ i64 │
+    ╞═════╪═════╡
+    │ xx  ┆ 2   │
+    │ xy  ┆ 4   │
+    │ yy  ┆ 5   │
+    │ yz  ┆ 6   │
+    └─────┴─────┘,
+    shape: (4, 2)
+    ┌─────┬─────┐
+    │ b1  ┆ b2  │
+    │ --- ┆ --- │
+    │ i64 ┆ i64 │
+    ╞═════╪═════╡
+    │ 3   ┆ 4   │
+    │ 5   ┆ 6   │
+    │ 6   ┆ 7   │
+    │ 7   ┆ 8   │
+    └─────┴─────┘]
+    >>> df.split.by_first_letter_of_column_values("a1")
+    [shape: (2, 4)
+    ┌─────┬─────┬─────┬─────┐
+    │ a1  ┆ a2  ┆ b1  ┆ b2  │
+    │ --- ┆ --- ┆ --- ┆ --- │
+    │ str ┆ i64 ┆ i64 ┆ i64 │
+    ╞═════╪═════╪═════╪═════╡
+    │ xx  ┆ 2   ┆ 3   ┆ 4   │
+    │ xy  ┆ 4   ┆ 5   ┆ 6   │
+    └─────┴─────┴─────┴─────┘, shape: (2, 4)
+    ┌─────┬─────┬─────┬─────┐
+    │ a1  ┆ a2  ┆ b1  ┆ b2  │
+    │ --- ┆ --- ┆ --- ┆ --- │
+    │ str ┆ i64 ┆ i64 ┆ i64 │
+    ╞═════╪═════╪═════╪═════╡
+    │ yy  ┆ 5   ┆ 6   ┆ 7   │
+    │ yz  ┆ 6   ┆ 7   ┆ 8   │
+    └─────┴─────┴─────┴─────┘]
+    """
+    return _create_namespace(name, nw.DataFrame)
+
+
+def register_lazyframe_namespace(name: str) -> Callable[[type[NS]], type[NS]]:
+    """Decorator for registering custom functionality with a Narwhals LazyFrame.
+
+    Parameters
+    ----------
+    name
+        Name under which the functionality will be accessed.
+
+    See Also:
+    --------
+    register_expr_namespace : Register functionality on an Expr.
+    register_dataframe_namespace : Register functionality on a DataFrame.
+    register_series_namespace : Register functionality on a Series.
+
+    Examples:
+    --------
+    >>> @nw.api.register_lazyframe_namespace("types")
+    ... class DTypeOperations:
+    ...     def __init__(self, lf: nw.LazyFrame) -> None:
+    ...         self._lf = lf
+    ...
+    ...     def split_by_column_dtypes(self) -> list[nw.LazyFrame]:
+    ...         return [
+    ...             self._lf.select(nw.col(tp))
+    ...             for tp in dict.fromkeys(self._lf.collect_schema().dtypes())
+    ...         ]
+    ...
+    ...     def upcast_integer_types(self) -> nw.LazyFrame:
+    ...         return self._lf.with_columns(
+    ...             nw.col(tp).cast(nw.Int64) for tp in (nw.Int8, nw.Int16, nw.Int32)
+    ...         )
+    >>>
+    >>> lf = nw.LazyFrame(
+    ...     data={"a": [1, 2], "b": [3, 4], "c": [5.6, 6.7]},
+    ...     schema=[("a", nw.Int16), ("b", nw.Int32), ("c", nw.Float32)],
+    ... )
+    >>> lf.collect()
+    shape: (2, 3)
+    ┌─────┬─────┬─────┐
+    │ a   ┆ b   ┆ c   │
+    │ --- ┆ --- ┆ --- │
+    │ i16 ┆ i32 ┆ f32 │
+    ╞═════╪═════╪═════╡
+    │ 1   ┆ 3   ┆ 5.6 │
+    │ 2   ┆ 4   ┆ 6.7 │
+    └─────┴─────┴─────┘
+    >>> lf.types.upcast_integer_types().collect()
+    shape: (2, 3)
+    ┌─────┬─────┬─────┐
+    │ a   ┆ b   ┆ c   │
+    │ --- ┆ --- ┆ --- │
+    │ i64 ┆ i64 ┆ f32 │
+    ╞═════╪═════╪═════╡
+    │ 1   ┆ 3   ┆ 5.6 │
+    │ 2   ┆ 4   ┆ 6.7 │
+    └─────┴─────┴─────┘
+
+    >>> lf = nw.LazyFrame(
+    ...     data=[["xx", 2, 3, 4], ["xy", 4, 5, 6], ["yy", 5, 6, 7], ["yz", 6, 7, 8]],
+    ...     schema=["a1", "a2", "b1", "b2"],
+    ...     orient="row",
+    ... )
+    >>> lf.collect()
+    shape: (4, 4)
+    ┌─────┬─────┬─────┬─────┐
+    │ a1  ┆ a2  ┆ b1  ┆ b2  │
+    │ --- ┆ --- ┆ --- ┆ --- │
+    │ str ┆ i64 ┆ i64 ┆ i64 │
+    ╞═════╪═════╪═════╪═════╡
+    │ xx  ┆ 2   ┆ 3   ┆ 4   │
+    │ xy  ┆ 4   ┆ 5   ┆ 6   │
+    │ yy  ┆ 5   ┆ 6   ┆ 7   │
+    │ yz  ┆ 6   ┆ 7   ┆ 8   │
+    └─────┴─────┴─────┴─────┘
+    >>> nw.collect_all(lf.types.split_by_column_dtypes())
+    [shape: (4, 1)
+    ┌─────┐
+    │ a1  │
+    │ --- │
+    │ str │
+    ╞═════╡
+    │ xx  │
+    │ xy  │
+    │ yy  │
+    │ yz  │
+    └─────┘, shape: (4, 3)
+    ┌─────┬─────┬─────┐
+    │ a2  ┆ b1  ┆ b2  │
+    │ --- ┆ --- ┆ --- │
+    │ i64 ┆ i64 ┆ i64 │
+    ╞═════╪═════╪═════╡
+    │ 2   ┆ 3   ┆ 4   │
+    │ 4   ┆ 5   ┆ 6   │
+    │ 5   ┆ 6   ┆ 7   │
+    │ 6   ┆ 7   ┆ 8   │
+    └─────┴─────┴─────┘]
+    """
+    return _create_namespace(name, nw.LazyFrame)
+
+
+def register_series_namespace(name: str) -> Callable[[type[NS]], type[NS]]:
+    """Decorator for registering custom functionality with a Narwhals Series.
+
+    Parameters
+    ----------
+    name
+        Name under which the functionality will be accessed.
+
+    See Also:
+    --------
+    register_expr_namespace : Register functionality on an Expr.
+    register_dataframe_namespace : Register functionality on a DataFrame.
+    register_lazyframe_namespace : Register functionality on a LazyFrame.
+
+    Examples:
+    --------
+    >>> @nw.api.register_series_namespace("math")
+    ... class MathShortcuts:
+    ...     def __init__(self, s: nw.Series) -> None:
+    ...         self._s = s
+    ...
+    ...     def square(self) -> nw.Series:
+    ...         return self._s * self._s
+    ...
+    ...     def cube(self) -> nw.Series:
+    ...         return self._s * self._s * self._s
+    >>>
+    >>> s = nw.Series("n", [1.5, 31.0, 42.0, 64.5])
+    >>> s.math.square().alias("s^2")
+    shape: (4,)
+    Series: 's^2' [f64]
+    [
+        2.25
+        961.0
+        1764.0
+        4160.25
+    ]
+    >>> s = nw.Series("n", [1, 2, 3, 4, 5])
+    >>> s.math.cube().alias("s^3")
+    shape: (5,)
+    Series: 's^3' [i64]
+    [
+        1
+        8
+        27
+        64
+        125
+    ]
+    """
+    return _create_namespace(name, nw.Series)
diff --git a/narwhals/dataframe.py b/narwhals/dataframe.py
index 303ff0636d..e8b13f2036 100644
--- a/narwhals/dataframe.py
+++ b/narwhals/dataframe.py
@@ -6,6 +6,7 @@
     TYPE_CHECKING,
     Any,
     Callable,
+    ClassVar,
     Generic,
     Literal,
     NoReturn,
@@ -425,6 +426,8 @@ class DataFrame(BaseFrame[DataFrameT]):
             ```
     """
 
+    _accessors: ClassVar[set[str]] = set()
+
     def _extract_compliant(self, arg: Any) -> Any:
         from narwhals.expr import Expr
         from narwhals.series import Series
@@ -2181,6 +2184,8 @@ class LazyFrame(BaseFrame[FrameT]):
         ```
     """
 
+    _accessors: ClassVar[set[str]] = set()
+
     def _extract_compliant(self, arg: Any) -> Any:
         from narwhals.expr import Expr
         from narwhals.series import Series
diff --git a/narwhals/expr.py b/narwhals/expr.py
index b5e01a3919..3bdf12212b 100644
--- a/narwhals/expr.py
+++ b/narwhals/expr.py
@@ -2,7 +2,7 @@
 
 import math
 from collections.abc import Iterable, Mapping, Sequence
-from typing import TYPE_CHECKING, Any, Callable
+from typing import TYPE_CHECKING, Any, Callable, ClassVar
 
 from narwhals._expression_parsing import (
     ExprMetadata,
@@ -53,6 +53,8 @@
 
 
 class Expr:
+    _accessors: ClassVar[set[str]] = {"cat", "dt", "list", "name", "str", "struct"}
+
     def __init__(self, to_compliant_expr: _ToCompliant, metadata: ExprMetadata) -> None:
         # callable from CompliantNamespace to CompliantExpr
         def func(plx: CompliantNamespace[Any, Any]) -> CompliantExpr[Any, Any]:
diff --git a/narwhals/series.py b/narwhals/series.py
index 147f7794ce..acc2ff9df9 100644
--- a/narwhals/series.py
+++ b/narwhals/series.py
@@ -2,7 +2,7 @@
 
 import math
 from collections.abc import Iterator, Mapping, Sequence
-from typing import TYPE_CHECKING, Any, Callable, Generic, Literal, overload
+from typing import TYPE_CHECKING, Any, Callable, ClassVar, Generic, Literal, overload
 
 from narwhals._utils import (
     _validate_rolling_arguments,
@@ -71,6 +71,8 @@ class Series(Generic[IntoSeriesT]):
             ```
     """
 
+    _accessors: ClassVar[set[str]] = {"cat", "dt", "list", "str", "struct"}
+
     @property
     def _dataframe(self) -> type[DataFrame[Any]]:
         from narwhals.dataframe import DataFrame