  We present a series of new results we obtained recently about the
intersection numbers of tautological classes on moduli spaces of curves,
including a simple formula of the n-point functions for Witten's $\tau$
classes, an effective recursion formula to compute higher Weil-Petersson
volumes, several new recursion formulae of intersection numbers and our proof
of a conjecture of Itzykson and Zuber concerning denominators of intersection
numbers. We also present Virasoro and KdV properties of generating functions of
general mixed $\kappa$ and $\psi$ intersections.
