  The higher-spin geometries of $W_\infty$-gravity and $W_N$-gravity are
analysed and used to derive the complete non-linear structure of the coupling
to matter and its symmetries. The symmetry group is a subgroup of the
symplectic diffeomorphisms of the cotangent bundle of the world-sheet, and the
$W_N$ geometry is obtained from a non-linear truncation of the $W_\infty$
geometry. Quantum W-gravity is briefly discussed. (Talk given at {\it Pathways
to Fundamental Interactions}, the 16th John Hopkins Workshop on Current
Problems in Particle Theory, Gothenborg, 1992.)
