  We propose that the Virasoro algebra controls quantum cohomologies of general
Fano manifolds $M$ ($c_1(M)>0$) and determines their partition functions at all
genera. We construct Virasoro operators in the case of complex projective
spaces and show that they reproduce the results of Kontsevich-Manin, Getzler
etc. on the genus-0,1 instanton numbers. We also construct Virasoro operators
for a wider class of Fano varieties. The central charge of the algebra is equal
to $\chi(M)$, the Euler characteristic of the manifold $M$.
