In this thesis, we investigate the period mapping of Teichmuller space into the Siegel upper half space. This is constructed from integrals of a basis of holomorphic one-forms along closed curves of a basis of the Riemann surface.  We consider the Riemann, Teichmuller and Torelli moduli spaces and their representation in the Siegel upper half space, and its relation to orbits of a symplectic and a set of positive polarizations of a vector space of dimension equal to the genus of the surface. / October 2016