Some Analytic and Topological Aspects of the Theory of Hodge Modules

This thesis concerns several applications of the analytic and topological theory of Saito's Hodge modules. The first part of the thesis gives an analytic proof of Saito's vanishing theorem by going back to the original idea of Kodaira. The second part concerns several natural Hodge modules on toric varieties, focusing on understanding the Hodge theoretic properties of the singularities of toric varieties. The last part of the thesis concerns various hypersurfaces that seem to be important for understanding the (compactified) period map defined on the GIT moduli space of Calabi--Yau type hypersurfaces.