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The theory of varieties for modules invokes algebraic geometry to illuminate representations of finite groups. Mathematicians have generalized the theory in a number of directions. In this talk we will give an overview in the context of finite dimensional Hopf algebras, which include group algebras, restricted Lie algebras, and small quantum groups, as well as many other algebras of current interest. We will discuss some applications, such as to representation type and to determining ideals in the tensor category of modules.