This is a survey of results on long time behavior and attractors for Hamiltonian nonlinear partial differential equations, considering the global attraction to stationary states, stationary orbits, and solitons, the adiabatic effective dynamics of the solitons, and the asymptotic stability of the solitary manifolds. The corresponding numerical results and relations to quantum postulates are considered. This theory differs significantly from the theory of attractors of dissipative systems where the attraction to stationary states is due to an energy dissipation caused by a friction. For the Hamilton equations the friction and energy dissipation are absent, and the attraction is caused by radiation which brings the energy irrevocably to infinity.