  (Talk given at Strings '93, Berkeley, and at XXVII. Internationales Symposium
\"uber Elementarteilchentheorie, Wendisch-Rietz, 1993) We review the
superconformal properties of matter coupled to $2d$ gravity, and $W$-extensions
thereof. We show in particular how the \nex2 structure provides a direct link
between certain matter-gravity systems and matrix models. We also show that
much, probably all, of this can be generalized to $W$-gravity, and this leads
to an infinite class of new exactly solvable systems. These systems are
governed by certain integrable hierarchies, which are generalizations of the
usual KdV hierarchy and whose algebraic structure is given in terms of quantum
cohomology rings of grassmannians.
