On Moduli Spaces of Calabi-Yau Manifolds

String theory is a ten-dimensional theory. If six of those dimensions are compactified on a specific type of manifold called Calabi-Yau, then at energies that are low compared to the Plank scale, a four-dimensional supergravity theory arises. This manifold possesses a certain number of parameters which span its moduli space. All of the questions raised in this Thesis concern this space. First, the issue of connectedness is addressed. We have proved that the moduli spaces of all Calabi-Yau manifolds, realized as single hypersurfaces in weighted projective spaces, are connected. This result helps to alleviate some of the discomfort that one feels that the Universe should have to select one of the myriad of Calabi-Yau compactifications. The complete structure of the moduli spaces of Calabi-Yau manifolds (and associated Landau-Ginzburg theories), and hence also of the corresponding low-energy effective theory that results from (2,2) superstring compactification, may be determined in terms of certain holomorphic functions called periods. The second part of this Thesis addresses this issue. An explicit computation of periods has been given, and it has been pointed out that it is possible to read off from them certain important information related to the mirror manifold. We also develop a homology basis that allows the simple calculation of periods as functions of the complex structure. The final issue addressed here concerns the building blocks for periods, called semi-periods. They are the solutions of the extended Picard-Fuchs equations before reduction of the order necessary to obtain periods. On the other hand, they are solutions of the integral of a three-form over the chain (instead of a cycle).