The time evolution of the one-particle distribution function of an N-particle classical Hamiltonian system with long-range interactions satisfies the Vlasov equation in the limit of infinite N. In this paper we present a new derivation of this result using a different approach allowing a discussion of the role of interparticle correlations on the system dynamics. Otherwise for finite N collisional corrections must be introduced. This has allowed a quite comprehensive study of the quasistationary states (QSSs) though many aspects of the physical interpretations of these states still remain unclear. In this paper a proper definition of time scale for long time evolution is discussed, and several numerical results are presented for different values of N. Previous reports indicate that the lifetimes of the QSS scale as N1.7 or even the system properties scale with exp(N). However, preliminary results presented here indicates that time scale goes as N2 for a different type of initial condition. We also discuss how the form of the interparticle potential determines the convergence of the N-particle dynamics to the Vlasov equation. The results are obtained in the context of the following models: the Hamiltonian mean field, the Self-gravitating ring model, and one- and two-dimensional systems of gravitating particles. We have also provided information of the validity of the Vlasov equation for finite N.