  From the cohomological point of view the symplectomorphism group $Sympl (M)$
of a symplectic manifold is `` tamer'' than the diffeomorphism group. The
existence of invariant polynomials in the Lie algebra $\frak {sympl }(M)$, the
symplectic Chern-Weil theory, and the existence of Chern-Simons-type secondary
classes are first manifestations of this principles. On a deeper level live
characteristic classes of symplectic actions in periodic cohomology and
symplectic Hodge decompositions. The present paper is called to introduce
theories and constructions listed above and to suggest numerous concrete
applications. These includes: nonvanishing results for cohomology of
symplectomorphism groups (as a topological space, as a topological group and as
a discrete group), symplectic rigidity of Chern classes, lower bounds for
volumes of Lagrangian isotopies, the subject started by Givental, Kleiner and
Oh, new characters for Torelli group and generalizations for automorphism
groups of one-relator groups, arithmetic properties of special values of Witten
zeta-function and solution of a conjecture of Brylinski. The Appendix, written
by L. Katzarkov, deals with fixed point sets of finite group actions in moduli
spaces.
