A review is presented of the one-parameter, nonsmooth bifurcations that occur in piecewise-smooth dynamical system. Motivated by applications, a pragmatic approach is taken to defining `bifurcation' as a nonsmooth transition with respect to a codimension-one discontinuity boundary in phase space. Only local bifurcations are considered, involving equilibria or a single point of boundary interaction along a limit cycle. Three classes of system are considered; involving either state jumps, jumps in the vector field, or in some derivative of the vector field. 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 identified, where possible, a normal form is given, together with a canonical example and an application. The goal is always to explain dynamics that may be observed in simulations.