  We generalize the local renormalization group (RG) equation to theories with
chiral anomalies. We find that a new anomaly is required by the Wess-Zumino
consistency conditions. Taking into account the new anomaly, the trace of the
energy momentum tensor is expressed in terms of the covariant flavor currents,
instead of the consistent ones. This result is used to show that a flavor
rotation induced by the RG flow can be eliminated by a choice of scheme even in
the presence of chiral anomalies. As part of a general discussion of chiral
anomalies in the presence of background sources, we also derive
non-renormalization theorems. Finally, we introduce the $\theta$ parameter as a
source, and derive constraints on a perturbative running of this parameter.
