There are many problems that lead to analysis of dynamical
systems in which one can distinguish motions of two types:
slow one and fast one. An averaging over fast motion is used
for approximate description of the slow motion. First integrals
of the averaged system are approximate first integrals of the
exact system, i.e. adiabatic invariants. Resonant phenomena
in fast motion (passage through separatrix, passage through
resonance, capture into resonance) lead to inapplicability
of averaging, destruction of adiabatic invariance and  dynamical
chaos in large domains in the phase space. In the talk some perturbation
theory methods for description of this phenomena are outlined.
As examples some problems from classical mechanics, hydrodynamics and
plasma physics are considered.