Quantum hypermultiplet moduli spaces in N = 2 string vacua: a review

The hypermultiplet moduli space MH in type II string theories compactified on a Calabi-Yau threefold X is largely constrained by supersymmetry (which demands quaternion- Kahlerity), S-duality (which requires an isometric action of SL(2,Z)) and regularity. Mathe- matically, MH ought to encode all generalized Donaldson-Thomas invariants on X consistently with wall-crossing, modularity and homological mirror symmetry. We review recent progress towards computing the exact metric on MH, or rather the exact complex contact structure on its twistor space.