I will introduce stable quasimaps to a certain GIT quotient W//G, to construct virtually smooth, moduli-theoretic, compactificatoin of maps from smooth projective curves to the smooth projective variety W//G; discuss possible applications. The induced theory should be conjecturally equivalent to GW theory for W//G. When W//G is a toric variety, the construction generalizes the toric compactification of Morrison, Plesser and Givental, allowing variations of the complex structures of domain curves. When W//G is a Grassmannian variety, the compactification coincides with the moduli of stable quotients recently introduced by Marian, Oprea and Pandharipande. This is joint work with I. Ciocan-Fontanine and D. Maulik.