  Lower-dimensional (hyper)surfaces that can carry gauge or gauge/gravitational
anomalies occur in many areas of physics: one-plus-one-dimensional boundaries
or two-dimensional defect surfaces in condensed matter systems,
four-dimensional brane-worlds in higher-dimensional cosmologies or various
branes and orbifold planes in string or M-theory. In all cases we may have
(quantum) anomalies localized on these hypersurfaces that are only cancelled by
``anomaly inflow'' from certain topological interactions in the bulk. Proper
cancellation between these anomaly contributions of different origin requires a
careful treatment of factors and signs. We review in some detail how these
contributions occur and discuss applications in condensed matter (Quantum Hall
Effect) and M-theory (five-branes and orbifold planes)
