Anomaly cancellation in M-theory: a critical review

We carefully review the basic examples of anomaly cancellation in M-theory: the 5-brane anomalies and the anomalies on S^1/Z_2. This involves cancellation between quantum anomalies and classical inflow from topological terms. To correctly fix all coefficients and signs, proper attention is paid to issues of orientation, chirality and the Euclidean continuation. Independent of the conventions chosen, the Chern-Simons and Green-Schwarz terms must always have the same sign. The reanalysis of the reduction to the heterotic string on S^1/Z_2 yields a surprise: a previously neglected factor forces us to slightly modify the Chern-Simons term, similar to what is needed for cancelling the normal bundle anomaly of the 5-brane. This modification leads to a local cancellation of the anomaly, while maintaining the periodicity on S^1.