  The w_\infty algebra is a particular generalization of the Virasoro algebra
with generators of higher spin 2,3,...,\infty. It can be viewed as the algebra
of a class of functions, relative to a Poisson bracket, on a suitably chosen
surface. Thus, w_\infty is a special case of area-preserving diffeomorphisms of
an arbitrary surface. We review various aspects of area- preserving
diffeomorphisms, w_\infty algebras and w_\infty gravity. The topics covered
include a) the structure of the algebra of area-preserving diffeomorphisms with
central extensions and their relation to w_\infty algebras, b) various
generalizations of w_\infty algebras, c) the structure of w_\infty gravity and
its geometrical aspects, d) nonlinear realizations of w_\infty symmetry and e)
various quantum realizations of w_\infty symmetry.
