Abstract: The focus of this talk will be on geometric properties of GW-invariants, especially of projective spaces and hypersurfaces, beginning with a geometric definition. My main objective is to compare and contrast the situation in genus 0 (morphisms from rational curves) with positive genus. In particular, I hope to indicate why the famous 1991 mirror symmetry prediction of Candelas-de la Ossa-Green-Parkes for genus 0 GW-invariants of a quintic threefold was verified within 5 years, while the 1993 genus 1 prediction of Bershadsky-Cecotti-Ooguri-Vafa took 14 years and the higher genus predictions are still open.