  We present the Constructive Methods of Invariant Manifolds for model
reduction in physical and chemical kinetics, developed during last two decades.
The problem of reduced description is studied as a problem of constructing the
slow invariant manifold. The invariance conditions are formulated as the
differential equation for a manifold immersed in the phase space (the
invariance equation). The equation of motion for immersed manifolds is obtained
(the film extension of the dynamics). Invariant manifolds are fixed points for
this equation, and slow invariant manifolds are Lyapunov stable fixed points,
thus slowness is presented as stability. A collection of methods for
construction of slow invariant manifolds is presented, in particular the
analogue of KAM methods for dissipative systems. We systematically consider a
discrete analogue of the slow (stable) positively invariant manifolds for
dissipative systems, invariant grids. The following examples of applications
are presented: Nonperturbative deviation of physically consistent hydrodynamics
from the Boltzmann equation and from the reversible dynamics, for Knudsen
numbers ~ 1; construction of the moment equations for nonequilibrium media and
their dynamical correction (instead of extension of list of variables) to gain
more accuracy in description of highly nonequilibrium flows; invariant grids
for chemical reactions; universal continuous media description of dilute
polymeric solution, etc.
