diff --git a/flox/core.py b/flox/core.py
index aa54f6757..4835da55b 100644
--- a/flox/core.py
+++ b/flox/core.py
@@ -420,7 +420,7 @@ def offset_labels(labels: np.ndarray, ngroups: int) -> tuple[np.ndarray, int]:
 def factorize_(
     by: tuple,
     axis: T_AxesOpt,
-    expected_groups: tuple[pd.Index, ...] = None,
+    expected_groups: tuple[pd.Index, ...] | None = None,
     reindex: bool = False,
     sort=True,
     fastpath=False,
@@ -873,7 +873,7 @@ def _simple_combine(
     return results
 
 
-def _conc2(x_chunk, key1, key2=slice(None), axis: T_Axes = None) -> np.ndarray:
+def _conc2(x_chunk, key1, key2=slice(None), axis: T_Axes | None = None) -> np.ndarray:
     """copied from dask.array.reductions.mean_combine"""
     from dask.array.core import _concatenate2
     from dask.utils import deepmap
@@ -1071,7 +1071,7 @@ def _reduce_blockwise(
     return result
 
 
-def _normalize_indexes(array, flatblocks, blkshape):
+def _normalize_indexes(array: DaskArray, flatblocks, blkshape) -> tuple:
     """
     .blocks accessor can only accept one iterable at a time,
     but can handle multiple slices.
@@ -1083,7 +1083,7 @@ def _normalize_indexes(array, flatblocks, blkshape):
     """
     unraveled = np.unravel_index(flatblocks, blkshape)
 
-    normalized: list[Union[int, np.ndarray, slice]] = []
+    normalized: list[Union[int, slice, list[int]]] = []
     for ax, idx in enumerate(unraveled):
         i = _unique(idx).squeeze()
         if i.ndim == 0:
@@ -1397,7 +1397,7 @@ def dask_groupby_agg(
     return (result, groups)
 
 
-def _collapse_blocks_along_axes(reduced, axis, group_chunks):
+def _collapse_blocks_along_axes(reduced: DaskArray, axis: T_Axes, group_chunks) -> DaskArray:
     import dask.array
     from dask.highlevelgraph import HighLevelGraph