Gromov-Witten invariants are a way to count maps from curves with specified genus and degree to a projective variety X. When X is a Calabi-Yau threefold, such as a quintic threefold, there are intriguing conjectures inspired from physics about the structure of the invariants.

In my talk, I will give an overview over the conjectures and the algebraic geometry behind the (now standard) computation of the genus zero invariants of quintic threefolds, and explain why it does not easily extend to higher genus. I will then proceed to discuss a construction (joint with Q. Chen and Y. Ruan) of new moduli spaces that can control the failure of the naive approach. In joint work with S. Guo and Y. Ruan, we use them to prove some of the conjecture about the structure of Gromov-Witten invariants of quintic threefolds.