  We prove that any three-point genus zero Gromov-Witten invariant on a type A
Grassmannian is equal to a classical intersection number on a two-step flag
variety. We also give symplectic and orthogonal analogues of this result; in
these cases the two-step flag variety is replaced by a sub-maximal isotropic
Grassmannian. Our theorems are applied, in type A, to formulate a conjectural
quantum Littlewood-Richardson rule, and in the other classical Lie types, to
obtain new proofs of the main structure theorems for the quantum cohomology of
Lagrangian and orthogonal Grassmannians.
