Weakly hoiomorphic forms play an important role in various areas of mathematics, the prominent example being the j-invariant. After reviewing the basic properties of holomorphic modular forms, we discuss the analogous statements for weakly holomorphic modular forms.  Then we extend the notion of periods of modular forms to the weakly holomorphic setting where they become quasiperiods. This leads in a natural way to a Hodge structure. Finally, we will look for these Hodge structures in geometry.