  Anomalies arising from nonplanar triangle diagrams of noncommutative gauge
theory are studied. Local chiral gauge anomalies for both noncommutative U(1)
and U(N) gauge theories with adjoint matter fields are shown to vanish. For
noncommutative QED with fundamental matters, due to UV/IR mixing a finite
anomaly emerges from the nonplanar contributions. It involves a generalized
$\star$-product of gauge fields.
