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I'll present recent results on the non-perturbative description of the hypermultiplet moduli space of compactified Type II string theory. The geometry of the moduli space gets instanton corrections due to D-branes and I'll show how one can describe all these D-instantons using the twistor approach to quaternionic geometries. 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 integrability.