In this talk, we give an introduction to the theory of period domains and period integrals. For any compact Kähler manifold X, the Hodge theorem gives a decomposition of the cohomology of X as the direct sum of complex vector spaces. Such a decomposition can be generalized to the notion of a Hodge structure, which has an equivalent description in terms of the Hodge filtration. Our goal is to study these variations on Hodge structures by viewing this filtration through the lens of the Grassmannian and flag manifolds. We provide a concrete example where we show that the parameter space for the polarized Hodge structures of a compact Riemann surface of genus q is analytically isomorphic to the Siegel upper half space.