Classifying Q-Fano 3-folds via mirror symmetry

The classification of Q-factorical terminal Fano three-folds is a long-standing open question. The possible Hilbert series were computed over 20 years ago by Reid et al., and in the intervening years approximately 100 cases have been shown to exist. Recent advances in Mirror Symmetry for Fano varieties, developed by Coates, Corti, and Kasprzyk, suggests a new approach: rather than considering the Fano varieties directly, instead consider their possible toric degenerations. In this thesis I will begin to sketch the classification of Q-factorical terminal Fano three-folds from this viewpoint. By exploiting new combinatorial techniques such as Laurent inversion, I will make significant contributions to the classification, constructing many explicit examples and significantly improving our understanding of these essential varieties.