Group cohomology is a powerful tool in group representation theory. To a group action on a vector space, one associates a geometric object called its support variety that is defined using group cohomology. Hopf algebras generalize groups and include many important classes of algebras such as Lie algebras and quantum groups. The theory of varieties for modules generalizes to Hopf algebras to some extent, but there are many open questions. In this introductory talk, we will define Hopf algebras, their cohomology, and the corresponding varieties for modules. We will discuss known and unknown properties and recent and current research on open problems