  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.
