  We survey the theory of attractors of nonlinear Hamiltonian partial
differential equations since its appearance in 1990. These are results on
global attraction to stationary states, to solitons and to stationary orbits,
on adiabatic effective dynamics of solitons and their asymptotic stability.
Results of numerical simulations are also given. Based on these results, we
propose a new general hypothesis on attractors of $G$-invariant nonlinear
Hamiltonian partial differential equations. The obtained results suggest a
novel dynamical interpretation of basic quantum phenomena: Bohr's transitions
between quantum stationary states, wave-particle duality, and probabilistic
interpretation.
