One of the fascinating aspects of string theory is that it lives in spacetime of ten dimensions. But this implies that, to relate it to phenomenological observations, it should be compactified down to four dimensions. A particularly rich, but still tractable case corresponds to compactifications on a Calabi-Yau manifold which gives at low energies an effective theory with N=2 supersymmetry. The action of this theory is completely determined by the metric on its moduli space which has two components corresponding to vector and hypermultiplets. The first is classically exact and well understood, whereas the latter receives quantum corrections and is known to carry a complicated quaternion-Kähler geometry. In this thesis we present our results on obtaining the complete non-perturbative description of the hypermultiplet moduli space. We show how all quantum corrections, which include perturbative one-loop contributions as well as non-perturbative ones due to D-brane and NS5-brane instantons, are incorporated in the framework of the twistor approach. This framework, which we elaborate here in detail, provides a powerful mathematical description of hyperkähler and quaternion-Kähler manifolds and is indispensable for formulating the non-perturbative geometry of the hypermultiplet moduli space. We also present new insights on S-duality, quantum mirror symmetry, connections to integrable models and topological strings.
