diff --git a/doc/source/development/extending.rst b/doc/source/development/extending.rst
index c7286616672b9..1d52a5595472b 100644
--- a/doc/source/development/extending.rst
+++ b/doc/source/development/extending.rst
@@ -488,3 +488,49 @@ registers the default "matplotlib" backend as follows.
 
 More information on how to implement a third-party plotting backend can be found at
 https://github.com/pandas-dev/pandas/blob/main/pandas/plotting/__init__.py#L1.
+
+.. _extending.pandas_priority:
+
+Arithmetic with 3rd party types
+-------------------------------
+
+In order to control how arithmetic works between a custom type and a pandas type,
+implement ``__pandas_priority__``.  Similar to numpy's ``__array_priority__``
+semantics, arithmetic methods on :class:`DataFrame`, :class:`Series`, and :class:`Index`
+objects will delegate to ``other``, if it has an attribute ``__pandas_priority__`` with a higher value.
+
+By default, pandas objects try to operate with other objects, even if they are not types known to pandas:
+
+.. code-block:: python
+
+    >>> pd.Series([1, 2]) + [10, 20]
+    0    11
+    1    22
+    dtype: int64
+
+In the example above, if ``[10, 20]`` was a custom type that can be understood as a list, pandas objects will still operate with it in the same way.
+
+In some cases, it is useful to delegate to the other type the operation. For example, consider I implement a
+custom list object, and I want the result of adding my custom list with a pandas :class:`Series` to be an instance of my list
+and not a :class:`Series` as seen in the previous example. This is now possible by defining the ``__pandas_priority__`` attribute
+of my custom list, and setting it to a higher value, than the priority of the pandas objects I want to operate with.
+
+The ``__pandas_priority__`` of :class:`DataFrame`, :class:`Series`, and :class:`Index` are ``4000``, ``3000``, and ``2000`` respectively.  The base ``ExtensionArray.__pandas_priority__`` is ``1000``.
+
+.. code-block:: python
+
+    class CustomList(list):
+        __pandas_priority__ = 5000
+
+        def __radd__(self, other):
+            # return `self` and not the addition for simplicity
+            return self
+
+    custom = CustomList()
+    series = pd.Series([1, 2, 3])
+
+    # Series refuses to add custom, since it's an unknown type with higher priority
+    assert series.__add__(custom) is NotImplemented
+
+    # This will cause the custom class `__radd__` being used instead
+    assert series + custom is custom
diff --git a/doc/source/whatsnew/v2.1.0.rst b/doc/source/whatsnew/v2.1.0.rst
index 7e8403c94ceef..ccf76f0fbc7fd 100644
--- a/doc/source/whatsnew/v2.1.0.rst
+++ b/doc/source/whatsnew/v2.1.0.rst
@@ -28,6 +28,7 @@ enhancement2
 
 Other enhancements
 ^^^^^^^^^^^^^^^^^^
+- Implemented ``__pandas_priority__`` to allow custom types to take precedence over :class:`DataFrame`, :class:`Series`, :class:`Index`, or :class:`ExtensionArray` for arithmetic operations, :ref:`see the developer guide <extending.pandas_priority>` (:issue:`48347`)
 - :meth:`MultiIndex.sort_values` now supports ``na_position`` (:issue:`51612`)
 - :meth:`MultiIndex.sortlevel` and :meth:`Index.sortlevel` gained a new keyword ``na_position`` (:issue:`51612`)
 - Improve error message when setting :class:`DataFrame` with wrong number of columns through :meth:`DataFrame.isetitem` (:issue:`51701`)
diff --git a/pandas/core/arrays/base.py b/pandas/core/arrays/base.py
index bbfe9b9bbb6c7..8c269244c37ce 100644
--- a/pandas/core/arrays/base.py
+++ b/pandas/core/arrays/base.py
@@ -235,6 +235,12 @@ class ExtensionArray:
     # Don't override this.
     _typ = "extension"
 
+    # similar to __array_priority__, positions ExtensionArray after Index,
+    #  Series, and DataFrame.  EA subclasses may override to choose which EA
+    #  subclass takes priority. If overriding, the value should always be
+    #  strictly less than 2000 to be below Index.__pandas_priority__.
+    __pandas_priority__ = 1000
+
     # ------------------------------------------------------------------------
     # Constructors
     # ------------------------------------------------------------------------
diff --git a/pandas/core/frame.py b/pandas/core/frame.py
index 70019030da182..bc28cc425a412 100644
--- a/pandas/core/frame.py
+++ b/pandas/core/frame.py
@@ -634,6 +634,10 @@ class DataFrame(NDFrame, OpsMixin):
     _hidden_attrs: frozenset[str] = NDFrame._hidden_attrs | frozenset([])
     _mgr: BlockManager | ArrayManager
 
+    # similar to __array_priority__, positions DataFrame before Series, Index,
+    #  and ExtensionArray.  Should NOT be overridden by subclasses.
+    __pandas_priority__ = 4000
+
     @property
     def _constructor(self) -> Callable[..., DataFrame]:
         return DataFrame
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 620823b9703ab..2f658006cf93f 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -351,6 +351,10 @@ class Index(IndexOpsMixin, PandasObject):
     # To hand over control to subclasses
     _join_precedence = 1
 
+    # similar to __array_priority__, positions Index after Series and DataFrame
+    #  but before ExtensionArray.  Should NOT be overridden by subclasses.
+    __pandas_priority__ = 2000
+
     # Cython methods; see github.com/cython/cython/issues/2647
     #  for why we need to wrap these instead of making them class attributes
     # Moreover, cython will choose the appropriate-dtyped sub-function
diff --git a/pandas/core/ops/common.py b/pandas/core/ops/common.py
index 01fb9aa17fc48..f8f53310e773b 100644
--- a/pandas/core/ops/common.py
+++ b/pandas/core/ops/common.py
@@ -14,7 +14,6 @@
 from pandas._libs.missing import is_matching_na
 
 from pandas.core.dtypes.generic import (
-    ABCDataFrame,
     ABCIndex,
     ABCSeries,
 )
@@ -75,10 +74,10 @@ def new_method(self, other):
             # For comparison ops, Index does *not* defer to Series
             pass
         else:
-            for cls in [ABCDataFrame, ABCSeries, ABCIndex]:
-                if isinstance(self, cls):
-                    break
-                if isinstance(other, cls):
+            prio = getattr(other, "__pandas_priority__", None)
+            if prio is not None:
+                if prio > self.__pandas_priority__:
+                    # e.g. other is DataFrame while self is Index/Series/EA
                     return NotImplemented
 
         other = item_from_zerodim(other)
diff --git a/pandas/core/series.py b/pandas/core/series.py
index e8d6491e43007..58cd42eaa7ca3 100644
--- a/pandas/core/series.py
+++ b/pandas/core/series.py
@@ -352,6 +352,10 @@ class Series(base.IndexOpsMixin, NDFrame):  # type: ignore[misc]
         base.IndexOpsMixin._hidden_attrs | NDFrame._hidden_attrs | frozenset([])
     )
 
+    # similar to __array_priority__, positions Series after DataFrame
+    #  but before Index and ExtensionArray.  Should NOT be overridden by subclasses.
+    __pandas_priority__ = 3000
+
     # Override cache_readonly bc Series is mutable
     # error: Incompatible types in assignment (expression has type "property",
     # base class "IndexOpsMixin" defined the type as "Callable[[IndexOpsMixin], bool]")
diff --git a/pandas/tests/test_downstream.py b/pandas/tests/test_downstream.py
index b17dce234043c..a97676578c079 100644
--- a/pandas/tests/test_downstream.py
+++ b/pandas/tests/test_downstream.py
@@ -267,3 +267,19 @@ def test_frame_setitem_dask_array_into_new_col():
         tm.assert_frame_equal(result, expected)
     finally:
         pd.set_option("compute.use_numexpr", olduse)
+
+
+def test_pandas_priority():
+    # GH#48347
+
+    class MyClass:
+        __pandas_priority__ = 5000
+
+        def __radd__(self, other):
+            return self
+
+    left = MyClass()
+    right = Series(range(3))
+
+    assert right.__add__(left) is NotImplemented
+    assert right + left is left