Mirror symmetry and quiver flag varieties

Quiver flag varieties are a generalization of type A flag varieties, first introduced by Alastair Craw. They are natural ambient spaces, and play a special role in the Fano classification program, where mirror constructions for these varieties are of particular interest. I'll survey some of the different mirror constructions available for Grassmannians - via toric degenerations, quantum cohomology, and the Abelian/non-Abelian correspondence - and discuss some of the work generalizing these constructions to quiver flag varieties.