  We use a topological framework to study descendent Gromov-Witten theory in
higher genus, non-toric settings. Two geometries are considered: surfaces of
general type and the Enriques Calabi-Yau threefold. We conjecture closed
formulas for surfaces of general type in classes K and 2K. For the Enriques
Calabi-Yau, Gromov-Witten invariants are calculated in genus 0, 1, and 2. In
genus 2, the holomorphic anomaly equation is found.
