To understand the dynamics of Hamiltonian systems, we need basic concepts like first integrals, integrability, special aspects of stability of periodic solutions and resonance. The use of the Poincaré recurrence theorem will play an important part to characterise the dynamics between regular and complex or chaotic. This is illustrated for the famous Hénon-Heiles system in the regular and in the chaotic domain. A systematic analysis of two- and three-dof systems shows interesting differences. For two-dof systems, the averaged-normal form is always integrable, and for three dof this is not the case. For two-dof systems we consider 1 : 2 and 1 : 1 resonances and higher order resonance; as an application, the spring pendulum is analysed at various resonances values. For three dof the first order resonances in normal form are given together with their integrability. The chapter closes with a discussion of adiabatic invariants and interaction of first and higher order resonance.