Modular forms and their higher-dimensional analogues are a priori analytically defined objects which happen to have many interesting relations to other subjects (such as number theory). In this lecture, I will review how algebraic geometry of modular curves (in mixed characteristics) was used for studying an important class of modular forms, and explain how geometry of the so-called Shimura varieties can be used for an analogous theory in higher dimensions. If time permits, I will also explain some interesting new application of such a theory to the study of torsion in the singular cohomology of Shimura varieties.