  We discuss how the theory of quantum cohomology may be generalized to
``gravitational quantum cohomology'' by studying topological sigma models
coupled to two-dimensional gravity. We first consider sigma models defined on a
general Fano manifold $M$ (manifold with a positive first Chern class) and
derive new recursion relations for its two point functions. We then derive
bi-Hamiltonian structures of the theories and show that they are completely
integrable at least at the level of genus $0$. We next consider the subspace of
the phase space where only a marginal perturbation (with a parameter $t$) is
turned on and construct Lax operators (superpotentials) $L$ whose residue
integrals reproduce correlation functions. In the case of $M=CP^N$ the Lax
operator is given by $L= Z_1+Z_2+\cdots +Z_N+e^tZ_1^{-1}Z_2^{-1}\cdots
Z_N^{-1}$ and agrees with the potential of the affine Toda theory of the $A_N$
type. We also obtain Lax operators for various Fano manifolds; Grassmannians,
rational surfaces etc. In these examples the number of variables of the Lax
operators is the same as the dimension of the original manifold. Our result
shows that Fano manifolds exhibit a new type of mirror phenomenon where mirror
partner is a non-compact Calabi-Yau manifold of the type of an algebraic torus
$C^{*N}$ equipped with a specific superpotential.
