Mirror symmetry is a major conjectured duality about Calabi-Yau manifolds, which provide the extra six dimensions of string theory, stating that families of Calabi-Yau manifolds have mirror partners with dual Hodge diamonds on which different versions of string theory would be physically indistinguishable. This may be formulated mathematically as a relationship between the Gromov-Witten theory of one member of a mirror pair and the solutions to Picard-Fuchs equations around the other. In the 1990s, Borcea-Voisin threefolds provided one of the first major classes of Calabi-Yau threefolds for which this conjecture was demonstrated at the Hodge diamond level, but at the Gromov-Witten level this was long poorly understood. A second compatible duality is the Landau Ginzburg/Calabi-Yau correspondence, which relates Calabi-Yau manifolds to systems arising from models of superconductivity, and has been recently generalised to a correspondence of gauged linear sigma models, often via certain intermediate theories. In this thesis we show both of these dualities hold for a range of Borcea-Voisin threefolds in the genus zero case. Speaker(s): Andrew Schaug (University of Michigan)