The Virasoro conjecture predicts that the generating
function of Gromov-Witten invariants is annihilated
by a sequence of differential operators which form
a half branch of the Virasoro algebra. This conjecture was
proposed by Eguchi-Hori-Xiong and S. katz. In the simplest
case where the underlying manifold is a point, this conjecture
is equivalent to Witten's KdV conjecture which was proved
by Kontsevich. In the talk I will describe some structures
of the Virasoro conjecture and the current status of this
conjecture up to genus 2.