The monograph is devoted to the study of nonlinear first order systems in the plane where the principal term is the gradient of a positive and positively 2-homogeneous Hamiltonian (or the convex combination of two of such gradients). After some preliminaries about positively 2-homogeneous autonomous systems, some results of existence and multiplicity of T-periodic solutions are presented in case of bounded or sublinear nonlinear perturbations. Our attention is mainly focused on the occurrence of resonance phenomena, and the corresponding results rely essentially on conditions of Landesman-Lazer or Ahmad-Lazer-Paul type. The techniques used are predominantly topological, exploiting the theory of coincidence degree and the use of the Poincar\ue9-Birkhoff fixed point theorem. At the end, other boundary conditions, including the Sturm-Liouville ones, are taken into account, giving the corresponding existence and multiplicity results in a nonresonant situation via the shooting method and topological arguments
