  We give a complete solution for the reduced Gromov-Witten theory of resolved
surface singularities of type A_n, for any genus, with arbitrary descendent
insertions. We also present a partial evaluation of the T-equivariant relative
Gromov-Witten theory of the threefold A_n x P^1 which, under a nondegeneracy
hypothesis, yields a complete solution for the theory. The results given here
allow comparison of this theory with the quantum cohomology of the Hilbert
scheme of points on the A_n surfaces. We discuss generalizations to linear
Hodge insertions and to surface resolutions of type D,E. As a corollary, we
present a new derivation of the stationary Gromov-Witten theory of P^1.
