This thesis develops a new approach to computing the quantum cohomology of symplectic reductions of partial flag varieties $X$; such symplectic reductions are known as weight varieties. Motivated by a conjecture of Teleman \cite{Tel14}, we use a mirror family Landau-Ginzburg model $(M_{P},f_{P})$ of $X$ introduced by Rietsch \cite{Rie08} to give a conjectural explicit description of the quantum cohomology of weight varieties.

We specialise to the class of polygon spaces $\mcP_{r,n}$, these are symplectic reductions of the complex Grassmannian of $2$-planes $\Gr_{\C}(2,n)$ by the maximal torus action. Polygon spaces in low rank have been classified and the quantum cohomology of these varieties is known. As a result, we are able to verify our conjectural description explicitly.

In addition, we investigate the appearance of representation-theoretic combinatorial structures in the mirror symmetry of complete flag varieties. We show that, on the $B$-model side, the extended string cone $\underline{C}_{\rexi}$ introduced by Caldero \cite{Cal02} to define toric degenerations on the $A$-model can be recovered via a discretisation process known as tropicalisation. Specifically, using a non-standard parameterisation of $M_{B}$, tropicalisation recovers the precise inequalities defining $\underline{C}_{\rexi}$. This provides an explicit approach to results previously obtained by Berenstein-Kazhdan \cite{BK07}. We conclude with a description of a conjectural program relating these combinatorial structures on the $B$-model with hierarchies of integrable systems on the $A$-model.