The aim of the paper is to give an overview of bifurcation phenomena which are typical for non-smooth dynamical systems. A small number of well-chosen examples of various kinds of non-smooth systems will be presented, followed by a discussion of the bifurcation phenomena in hand and a brief introduction to the mathematical tools which have been developed to study these phenomena. The bifurcations of equilibria in two planar non-smooth continuous systems are analysed by using a generalised Jacobian matrix. A mechanical example of a non-autonomous Filippov system, belonging to the class of differential inclusions, is studied and shows a number of remarkable discontinuous bifurcations of periodic solutions. A generalisation of the Floquet theory is introduced which explains bifurcation phenomena in differential inclusions. Lastly, the dynamics of the Woodpecker Toy is analysed with a one-dimensional Poincaré map method. The dynamics is greatly influenced by simultaneous impacts which cause discontinuous bifurcations.