  We state and prove a topological recursion relation that expresses any
genus-g Gromov-Witten invariant of a projective manifold with at least a
(3g-1)-st power of a cotangent line class in terms of invariants with fewer
cotangent line classes. For projective spaces, we prove that these relations
together with the Virasoro conditions are sufficient to calculate the full
Gromov-Witten potential. This gives the first computationally feasible way to
determine the higher genus Gromov-Witten invariants of projective spaces.
