Abstract: The averaging method gives a quantitative relation between the solutions of some nonautonomous differential system and the solutions of its autonomous averaged differential system, and in particular allows to study the periodic orbits of the nonautonomous differential system in function of the periodic orbits of the averaged one. In this talk we will apply the averaging theory to two classes of Hamiltonian systems and also to some specific classes of differential equations for studying their periodic solutions. (This is joint work with J. Llibre).