Representations of vertex operator algebras

This paper is an exposition of the representation theory of vertex operator algebras in terms of associative algebras A n (V ) and their bimodules.A new result on the rationality is given.That is, a simple vertex operator algebra V is rational if and only if its Zhu algebra A(V ) is a semisimple associative algebra and each irreducible admissible V -module is ordinary.2000MSC:17B69