Introduction

The Conway-Norton conjectures about the Monster-modular connection, later established by Borcherds [B] following the work of Frenkel-Lepowsky-Meurman [FLM], set the stage for an intensive study of the origins of the relations between finite groups and modular functions. Norton also introduced generalized moonshine which associates q-expansions to pairs of commuting elements in M. His conjecture that the nonconstant functions that one obtains in this way are also hauptmoduln remains open.

By now it is clear that the principal mathematical idea underlying the general study of such phenomena is that of a vertex operator algebra. For an extensive class of vertex operator algebras, so-called rational orbifold models, one expects a theory completely parallel to Monstrous Moonshine whereby one associates modular functions to automorphisms of finite order. Furthermore, this theory naturally accommodates generalized moonshine. From this perspective, the Conway-Norton phenomena would be a particularly interesting spradic example of a general theory, just as the Monster itself is a particularly interesting sporadic example of a finite simple group.

In the spirit of the Edinburgh Conference, this paper is mainly devoted to a review of some of the main results currently available concerning the structure of rational vertex operator algebras and their orbifolds. We sometimes use the Frenkel-Lepowsky-Meurman Moonshine Module V♭ [FLM] to illustrate the ideas. We also suggest open problems - many of them well-known - whose solution would contribute to a more complete theory.