Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of representation theoretic vector bundles. Grassmannians are an example of quiver flag varieties. The Abelian/non-Abelian correspondence is a conjecture relating the Gromov--Witten invariants of a non-Abelian GIT quotient to the same invariants of an Abelian GIT quotient. In the first chapter, we show how the conjecture in the case of Grassmannians arises from Givental's loop space mirror heuristics. We then prove the Abelian/non-Abelian Correspondence for quiver flag zero loci: this allows us to compute their genus zero Gromov--Witten invariants. We determine the ample cone of a quiver flag variety. In joint work with Tom Coates and Alexander Kasprzyk, we use these results to find all four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum periods. In this way we find at least 141 new four-dimensional Fano manifolds. In the last chapter, we describe a conjectural method for finding mirrors to these fourfolds, and implement this in several examples.