Résumé
I'll review the twistor approach to the description of hyperkahler and quaternion-Kahler geometries allowing their efficient parametrization in terms of a set of holomorphic potentials. These geometries appear as moduli spaces in N=2 supersymmetric gauge theories and in compactifications of Type II superstring theories on Calabi-Yau threefolds. In particular, using the twistor approach, I'll present a non-perturbative description of the hypermultiplet moduli space, which determines the low energy effective action of compactified string theory. The geometry of this moduli space gets instanton corrections due to various branes wrapping non-trivial cycles of the Calabi-Yau. I'll show how one can describe all instantons coming from D-branes and tentative results on instanton contributions from NS5-branes. These results are used to get a quantum completion of the classical mirror map and to show a deep relation of the twistor description of D-instantons to integrable models. In particular, I'll show that the D-instanton corrections are described by Thermodynamic Bethe Ansatz and the corresponding S-matrix satisfies all axioms of integrability.