Mirror symmetry and the classification of orbifold del Pezzo surfaces

We state a number of conjectures that together allow one to classify a broad class of del Pezzo surfaces with cyclic quotient singularities using mirror symmetry. We prove our conjectures in the simplest cases. The conjectures relate mutation-equivalence classes of Fano polygons with $\mathbb {Q}$-Gorenstein deformation classes of del Pezzo surfaces.