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Superstring theories are the most promising theories for unified descriptionof all fundamental interactions including gravity. However, these theories areformulated consistently only in 10 spacetime dimensions. Therefore, to connectto the observable world, it is required to compactify 6 out of those 10dimensions in a suitable fashion.  In this thesis, we mainly consider compactifications of type II stringtheories on Calabi-Yau threefolds. As a consequence, the resulting fourdimensional theories preserve $\mathcal{N}=2$ supersymmetry. In these cases themetrics on the moduli spaces of the matter multiplets, vector andhypermultiplets, completely determine the low energy theories. Whereas theformer are very well understood by now, the complete description ofhypermultiplets is more complicated. In fact, hypermultiplets receive bothperturbative and non-perturbative corrections. The thesis mainly pertains tothe understanding of the non-perturbative corrections. Our findings for thehypermultiplets rely on the so called twistorial construction. We discuss thistechnique in details and use it throughout the course of this thesis. Inparticular, armed with this, and exploiting various symmetries, especiallyS-duality and mirror symmetry, we discuss the procedure to derive the D-braneand NS5-brane instanton corrections to the hypermultiplet moduli space.

Calabi-Yau compactification of type II string theories