In this very insightful survey article, the author describes results on long time behavior and attractors for nonlinear Hamiltonian systems. In the introduction, the beginnings of the mathematical theory of attractors for nonlinear PDEs and important developments are considered, with emphasis for the Hamiltonian PDEs, where the energy radiation to infinity plays the role of dissipation, in the attraction to proper attractors. First the global attractors to stationary states is considered (in Section 2), then the global attractors to stationary orbits (Section 3) and to solitons (Section 4) are considered. Section 5 covers the adiabatic effective dynamics of solitons. In the key Section 6, the asymptotic stability of solitary waves is considered. Here we quote the author: ``In a series of papers an ingenious strategy was developed for proving the asymptotic stability of the trajectory onto the solitary manifold. In particular, this strategy includes the symplectic projection of the trajectory onto the solitary manifold, the modulation equations for the soliton parameters of the projection, and the decay of the transversal component. This approach is a far-reaching development of the Lyapunov stability theory.'' In Subsection 6.1, the linearization and decomposition of the dynamics is described. Subsection 6.2 describes the cunning strategy adopted in the references [\textit{V. S. Buslaev} and \textit{G. S. Perel'man}, St. Petersbg. Math. J. 4, No. 6, 63--102 (1992; Zbl 0853.35112); translation from Algebra Anal. 4, No. 6, 63--102 (1992); in: Nonlinear evolution equations. Providence, RI: American Mathematical Society. 75--98 (1995; Zbl 0841.35108); \textit{V. S. Buslaev} and \textit{C. Sulem}, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20, No. 3, 419--475 (2003; Zbl 1028.35139)] to obtain the asymptotics. Subsection 6.3 describes extensions and applications, while in Subsection 6.4, further results (after 2003) in several directions are described. In Subsection 6.5, recent results on dispersion decay of solution to the corresponding linearized equation are described. The paper ends with Section 7 on numerical simulations of soliton asymptotics, where the joint work with A. Vinnichenko is described. Numerical experiments are shown in agreement with the radiation mechanism described at Subsection 3.8 (on disperson radiation and nonlinear energy transfer): (i) The nonlinearity causes the energy transfer from the lower to the higher modes, (ii) then, the dispersion radiation of the higher modes transports their energy to infinity. The numerical experiments show the same effect for nonlinear relativistically-invariant wave equations. The paper ends with an Appendix on attractors and quantum postulates