This paper treats bifurcations of periodic solutions in discontinuous systems of the Filippov type. Furthermore, bifurcations of fixed points in non-smooth continuous systems are addressed.Filippov's theory for the definition of solutions of discontinuous systems is surveyed and jumps infundamental solution matrices are discussed. It is shown how jumps in the fundamental solution matrix lead to jumps of the Floquet multipliers of periodic solutions. The Floquet multipliers can jump through the unit circle causing discontinuous bifurcations. Numerical examples are treated which show variousdiscontinuous bifurcations. Also infinitely unstable periodic solutions are addressed.