  We work out the one-loop $U(1)_A$ anomaly for noncommutative SU(N) gauge
theories up to second order in the noncommutative parameter $\theta^{\mu\nu}$.
We set $\theta^{0i}=0$ and conclude that there is no breaking of the classical
$U(1)_A$ symmetry of the theory coming from the contributions that are either
linear or quadratic in $\theta^{\mu\nu}$. Of course, the ordinary anomalous
contributions will be still with us. We also show that the one-loop
conservation of the nonsinglet currents holds at least up to second order in
$\theta^{\mu\nu}$. We adapt our results to noncommutative gauge theories with
SO(N) and U(1) gauge groups.
