The dynamics of nearly integrable Hamiltonian is called by Poincaré the fundamental problem of dynamical systems. Since the fifties of the last century, the great achievements have been made in the study of the problem. Among them, the Kolmogorov's theorem on invariant tori and the discovery of Arnold diffusion are the landmarks. They have greatly deepened our understanding on the dynamical diversity of nearly integrable Hamiltonian systems. We shall briefly review some of the developments.