We revisit quantum field theory anomalies, emphasizing the interplay with diffeomorphisms and supersymmetry. The Ward identities of the latter induce Noether currents of all continuous symmetries, and we point out how these consistent currents are replaced by their covariant form through the appearance of the Bardeen-Zumino currents, which play a central role in our study. For supersymmetry Ward identities, two systematic methods for solving the Wess-Zumino consistency conditions are discussed: anomaly inflow and anomaly descent. The simplest inflows are from supersymmetric Chern-Simons actions in one dimension higher, which are used to supersymmetrize flavor anomalies in $d=4$ and, for $d=2$ $\mathcal{N}=(p,q)$, flavor anomalies with $p,q\leq 3$ and Lorentz-Weyl anomalies with $p,q\leq 6$. Finally, we extend the BRST algebra and the subsequent descent, a necessity for the diffeomorphism anomaly in retrospect. The same modification computes the supersymmetrized anomalies, and determines the above Chern-Simons actions when these exist.

65 pages; v2: fixed minor typos and added references. Version to appear in PRD