In this thesis we study the enumerative geometry of certain GIT quotients. Chapter 2 details work (joint with T. Coates and W. Lutz) on the comparison between genus-zero Gromov–Witten invariants of X and of a blow up of X. We do this by rewriting a blow up as a subvariety of a certain GIT quotient and extending the Abelian/non-Abelian correspondence, which compares genus-zero Gromov–Witten invariants of a GIT quotient by a non-abelian group with the corresponding quotient by the maximal torus. We also give a reformulation of the Abelian/non-Abelian correspondence in terms of Givental’s Lagrangian cones, which suggests a relationship in higher genus. In Chapter 3 we build a theory of logarithmic quasimaps for smooth projective toric varieties relative a simple normal crossings divisor in any genus. To do this we construct a proper Deligne–Mumford moduli stack compactifying the space of quasimaps from smooth marked curves of fixed degree and genus satisfying prescribed tangency conditions to the divisor at the markings. We give a sketch of the construction of the virtual fundamental class using the formalism of Behrend and Fantechi. We then look at how this theory interacts with the local-logarithmic correspondence.