A review is presented of the one-parameter, nonsmooth bifurcations that occur in a variety of continuous time piecewise-smooth dynamical systems. Motivated by applications, a pragmatic approach is taken to dening discontinuity-induced bifurcation (DIB) as a non-trivial interaction of a limit set with respect to a codimension-one discontinuity boundary in phase space. Only DIBs that are local are considered, that is bifurcations involving equilibria or a single point of boundary interaction along a limit cycle for ows. Three classes of systems are considered; involving either state jumps, jumps in the vector eld, or in some derivative of the vector eld. A rich array of dynamics are revealed, involving the sudden creation or disappearance of attractors, jumps to chaos, bifurcation diagrams with sharp corners, and cascades of period adding. For each kind of bifurcation identied, where possible, a kind of `normal form' or discontinuity mapping (DM) is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations of systems which include friction oscillates, impact oscillators, DC-DC converters and problems in control theory.