As explained by Auroux, the Strominger--Yau--Zaslow heuristic predicts that the mirror of a Fano $n$-fold is a family of Calabi--Yau $(n-1)$-folds over the affine line with maximally unipotent monodromy at infinity. We first explain how this prediction together with work of Doran--Harder--Novoseltsev--Thompson recovers the classification of Fano $3$-folds with $b_2=1$. Due to the minimal model program, the classification of $b_2=1$ Fano 3-folds with so-called terminal singularities is an important open problem. Miles Reid et al described a list of approximately $40,000$ candidate Hilbert series for such ${\mathbb Q}$-Fano $3$-folds, most of which are not known to exist. They also classified examples of codimension $\le 3$ in the anti-canonical embedding (for example, there are $95$ deformation types of ${\mathbb Q}$-Fano hypersurfaces in weighted projective space). We explain how to modify the above heuristic in the ${\mathbb Q}$-Fano case and discuss applications to classification.

This is joint work in progress with Alessio Corti.