One of the main open questions in string theory is understanding how to characterize generic string compactifications. The answer to this question has far-reaching implications on our ability to extract physical predictions from string theory and connect it to the observable universe. While a great deal of work has been done on compactifying string theory on specific backgrounds, such as Calabi-Yau spaces, a general picture of the space of compactifications remains elusive. In this work, we explore the parameter space of gauged linear sigma models (GLSM) with (2,2) and (0,2) supersymmetry and study the geometries that they can describe. First we provide a more general picture of the essential couplings that should be considered in a (2,2) GLSM. We then explore analogues of (2,2) quantum cohomology rings for (0,2) theories which are not obtainable as deformations of theories on the (2,2) locus. Finally, we explore abelian duality of chiral (0,2) GLSM. The duals of these models generically involve charged fields, unlike those of their (2,2) counterparts.