In this chapter we prove the famous Poincaré–Bendixson Theorem that classifies the asymptotic behaviour of bounded orbits in smooth planar ODEs. We shall see that in a generic planar system any such orbit tends either to an equilibrium or to a cycle (periodic orbit), although in general more complicated limit sets are possible. Then we give some conditions ensuring existence or nonexistence of cycles in planar systems, in particular, when the system is close to a special (Hamiltonian) system. We apply many results and techniques to the analysis of several prey–predator ecological models. Basic facts about multidimensional Hamiltonian systems are collected in the Appendix.