H I S T O R I C A L N O T E
This paper was distributed as a preprint starting in 1989. Except for a few
minor corrections, it is identical to the original preprint. The paper develops the
basic axiomatic theory of vertex operator algebras; some of this material was
already incorporated in the book "Vertex operator algebras and the Monster"
[FLM] (1988), as was indicated there. Since then, these ideas have been applied
in several directions by a number of people, and many new examples of vertex
operator algebras and related structures have been studied. The importance of
the category of vertex operator algebras has become more evident. We believe
that the publication of this paper at the present time is no less useful than when
it was written.