diff --git a/distributed/dashboard/tests/test_scheduler_bokeh.py b/distributed/dashboard/tests/test_scheduler_bokeh.py
index 80b0ba29166..909b122c062 100644
--- a/distributed/dashboard/tests/test_scheduler_bokeh.py
+++ b/distributed/dashboard/tests/test_scheduler_bokeh.py
@@ -1118,6 +1118,7 @@ async def test_shuffling(c, s, a, b):
     ss = Shuffling(s)
 
     df = dask.datasets.timeseries()
+    df["name"] = df["name"].astype("string[python]")
     df2 = dd.shuffle.shuffle(df, "x", shuffle="p2p").persist()
     start = time()
     while not ss.source.data["disk_read"]:
diff --git a/distributed/shuffle/_arrow.py b/distributed/shuffle/_arrow.py
index f0dc722ece3..bf1aae981ba 100644
--- a/distributed/shuffle/_arrow.py
+++ b/distributed/shuffle/_arrow.py
@@ -4,9 +4,30 @@
 from typing import TYPE_CHECKING, BinaryIO
 
 if TYPE_CHECKING:
+    import pandas as pd
     import pyarrow as pa
 
 
+def check_dtype_support(meta_input: pd.DataFrame) -> None:
+    import pandas as pd
+
+    for name in meta_input:
+        column = meta_input[name]
+        # FIXME: PyArrow does not support complex numbers: https://issues.apache.org/jira/browse/ARROW-638
+        if pd.api.types.is_complex_dtype(column):
+            raise TypeError(
+                f"p2p does not support data of type '{column.dtype}' found in column '{name}'."
+            )
+        # FIXME: Serializing custom objects to PyArrow is not supported in P2P shuffling
+        if pd.api.types.is_object_dtype(column):
+            raise TypeError(
+                f"p2p does not support custom objects found in column '{name}'."
+            )
+        # FIXME: PyArrow does not support sparse data: https://issues.apache.org/jira/browse/ARROW-8679
+        if pd.api.types.is_sparse(column):
+            raise TypeError("p2p does not support sparse data found in column '{name}'")
+
+
 def dump_shards(shards: list[bytes], file: BinaryIO) -> None:
     """
     Write multiple shard tables to the file
diff --git a/distributed/shuffle/_shuffle.py b/distributed/shuffle/_shuffle.py
index 8aa0adf7072..b1eee5bc052 100644
--- a/distributed/shuffle/_shuffle.py
+++ b/distributed/shuffle/_shuffle.py
@@ -7,6 +7,8 @@
 from dask.highlevelgraph import HighLevelGraph
 from dask.layers import SimpleShuffleLayer
 
+from distributed.shuffle._arrow import check_dtype_support
+
 logger = logging.getLogger("distributed.shuffle")
 if TYPE_CHECKING:
     import pandas as pd
@@ -94,6 +96,7 @@ def rearrange_by_column_p2p(
 ) -> DataFrame:
     from dask.dataframe import DataFrame
 
+    check_dtype_support(df._meta)
     npartitions = npartitions or df.npartitions
     token = tokenize(df, column, npartitions)
 
@@ -103,12 +106,6 @@ def rearrange_by_column_p2p(
         raise TypeError(
             f"p2p requires all column names to be str, found: {unsupported}",
         )
-    for c, dt in empty.dtypes.items():
-        if dt == object:
-            empty[c] = empty[c].astype(
-                "string"
-            )  # TODO: we fail at non-string object dtypes
-    empty[column] = empty[column].astype("int64")  # TODO: this shouldn't be necesssary
 
     name = f"shuffle-p2p-{token}"
     layer = P2PShuffleLayer(
diff --git a/distributed/shuffle/tests/test_graph.py b/distributed/shuffle/tests/test_graph.py
index 4f2d6add1fd..6f1299282f9 100644
--- a/distributed/shuffle/tests/test_graph.py
+++ b/distributed/shuffle/tests/test_graph.py
@@ -29,6 +29,38 @@ def test_basic(client):
     # ^ NOTE: this works because `assert_eq` sorts the rows before comparing
 
 
+@pytest.mark.parametrize("dtype", ["csingle", "cdouble", "clongdouble"])
+def test_raise_on_complex_numbers(dtype):
+    df = dd.from_pandas(
+        pd.DataFrame({"x": pd.array(range(10), dtype=dtype)}), npartitions=5
+    )
+    with pytest.raises(
+        TypeError, match=f"p2p does not support data of type '{df.x.dtype}'"
+    ):
+        df.shuffle("x", shuffle="p2p")
+
+
+def test_raise_on_custom_objects():
+    class Stub:
+        def __init__(self, value: int) -> None:
+            self.value = value
+
+    df = dd.from_pandas(
+        pd.DataFrame({"x": pd.array([Stub(i) for i in range(10)], dtype="object")}),
+        npartitions=5,
+    )
+    with pytest.raises(TypeError, match="p2p does not support custom objects"):
+        df.shuffle("x", shuffle="p2p")
+
+
+def test_raise_on_sparse_data():
+    df = dd.from_pandas(
+        pd.DataFrame({"x": pd.array(range(10), dtype="Sparse[float64]")}), npartitions=5
+    )
+    with pytest.raises(TypeError, match="p2p does not support sparse data"):
+        df.shuffle("x", shuffle="p2p")
+
+
 def test_raise_on_non_string_column_name():
     df = dd.from_pandas(pd.DataFrame({"a": range(10), 1: range(10)}), npartitions=5)
     with pytest.raises(TypeError, match="p2p requires all column names to be str"):
@@ -43,6 +75,7 @@ def test_does_not_raise_on_stringified_numeric_column_name():
 @gen_cluster([("", 2)] * 4, client=True)
 async def test_basic_state(c, s, *workers):
     df = dd.demo.make_timeseries(freq="15D", partition_freq="30D")
+    df["name"] = df["name"].astype("string[python]")
     shuffled = df.shuffle("id", shuffle="p2p")
 
     exts = [w.extensions["shuffle"] for w in workers]