[en] The w_∞ algebra is a particular generalization of the Virasoro algebra with generators of higher spin 2, 3, ... ∞. It can be viewed as the algebra of a class of functions, relative to a Poisson bracket, on a suitably chosen surface. Thus, w_∞ is a special case of area-preserving diffeomorphisms of an arbitrary surface. We review various aspects of area-preserving diffeomorphisms, w_∞ algebras and w_∞ gravity. The topics covered include a) the structure of the algebra of area-preserving diffeomorphisms with central extensions and their relation to w_∞ algebras, b) various generalizations of w_∞ algebras, c) the structure of w_∞ gravity and its geometrical aspects, d) nonlinear realizations of w_∞ symmetry and e) various quantum realizations of w_∞ symmetry. (author). 40 refs