String theory requires a ten-dimensional spacetime. In order to make contact with our four dimensional world, six of these dimensions have to be made compact and very small. The structure of the compact space governs the physics in four dimensions. For the construction of realistic string models it is essential to understand what happens in the extra dimensions. The main focus of my research is to learn about the mathematical structure of the compact dimensions, and to explore dualities, such as mirror symmetry, which relate different string compactifications.