To Vertex Operator Algebras

In this exposition, we continue the discussions of Dong [D2] and Li [L]. We shall prove an S3-symmetry of the Jacobi identity, construct the contragredient module for a module for a vertex operator algebra and apply these to the construction of the vertex operator map for the moonshine module. We shall introduce the notions of intertwining operator, fusion rule and Verlinde algebra. We shall also describe briefly the geometric interpretation of vertex operator algebras. We end the exposition with an explanation of the role of vertex operator algebras in conformal field theories. I would like to thank Masahiko Miyamoto for inviting me to this successful conference and James Lepowsky for helpful mathematical comments.