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Abstract: I will describe important topics in the representation theory of vertex operator algebras (VOAs). Under certain regularity assumptions the graded dimensions (or characters) of VOAs and of their irreducible modules are modular forms. As such they transform in a certain way under the action of SL_2(Z) on the complex upper half-plane. The miraculous Verlinde formula provides a relation between the image of a certain matrix S under this action and the fusion rules of the irreducible modules.