The Virasoro conjecture predicts that the generating function of Gromov-Witten invariants of any smooth projective variety is annihilated by an infinite sequence of differential operators which form a half branch of the Virasoro algebra. In case the underlying manifold is a point, this conjecture is equivalent to Witten's conjecture, proved by Kontsevich, that the generating function of intersection numbers on moduli spaces of stable cures is a tau function of the KdV hierarchy. In an earlier paper with G. Tian, we proved the genus-0 part of the Virasoro conjecture. In this talk I will discuss the current status of this conjecture for the genus-1 case.