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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

ANOMALY CANCELLATIONS ON LOWER-DIMENSIONAL HYPERSURFACES BY INFLOW FROM THE BULK