$\bf W_\infty$ Gravity - a Geometric Approach

A brief review is given of an adaptation of the coadjoint orbit method appropriate for study of models with infinite-dimensional symmetry groups. It is illustrated on several examples, including derivation of the WZNW action of induced $D=2\,$ $(N,0)\,$ supergravity. As a main application, we present the geometric action on a generic coadjoint orbit of the deformed group of area preserving diffeomorphisms. This action is precisely the anomalous effective WZNW action of $D=2 \,$ matter fields coupled to chiral $W_\infty$ gravity background. Similar actions are given which produce the {\em KP} hierarchy as on-shell equations of motion.