Diffeomorphism anomalies differ from gauge anomalies, since the relevant transformations start with translations. The subtlety is often side-stepped by resorting to local Lorentz anomalies, yet the latter does not measure the failure of the gravitational Ward identity. We devote the first third of the chapter to a detailed study of diffeomorphisms and the anomalies thereof, including a new twist; the diffeomorphism generator on spinors used in most physics literature is incomplete and must be replaced by the Kosmann lift. This leads to a factor 1/2 correction to the legacy computation of the covariant anomaly and restores the correct normalization of the consistent anomaly that has been in use without real justification. We then move on to other key facts and machineries of anomalies, such as anomaly matching and anomaly inflow, that are of more modern relevance. We also devote a substantial part of the chapter to explicit forms of anomaly polynomials and lay out important relations among them that originate from Lie algebra representation theory.