So far in this book we mainly concentrated on the (changes in) local behaviour near steady states (including periodic orbits of maps and, in principle but not so easy in practice, periodic orbits of ODEs via the Poincaré map). The idea may arise that whenever orbits stay bounded the persistent behaviour is either constant in time or shows regular oscillations. The main aim of this chapter is to show that this idea is wrong: More complicated yet persistent behaviour of dynamical systems is possible and does occur. The main topics are In Appendices we will discuss general results about the bifurcation of periodic orbits from a homoclinic orbit in n-dimensional ODEs and the existence of complicated invariant sets of n-dimensional maps near their transverse homoclinic orbits.