Preface One of the great legacies of the classification of the finite simple groups is the existence of the Monster. It was the study of this group that first suggested that there might be interesting relations between finite groups and certain ellip- tic modular functions, and it was this possibility-fuelled by the Conway-Norton conjectures-that led to what one might call the first version of "Moonshine", that is, the study of class functions on groups with values in a ring of modular functions. Work of Borcherds and Frenkel-Lepowsky-Meurman led to the notion of aver- tex (operator) algebra, which was seen to be the same as the chiral algebras used by physicists in conformal field theory. Nowadays one considers the Monster as the group of automorphisrns of a certain vertex operator algebra-the so-called Moon- shine Module-and "Moonshine" may be construed as the representation theory of certain vertex operator algebras, namely so-called orbifolds. The connections with physics have proven to be invaluable, and it seems likely that another branch of mathematics whose origins are eerily similar to those of moonshine-that is, elliptic cohomology-will turn out to be very relevant too. Most of the talks at the Moonshine Conference were devoted to one or more of these subjects, as are the accompanying papers in this volume. Way back in 1980 in the Proceedings of the Santa Cruz conference on finite groups, when Moonshine was just emerging, Andy Ogg wrote, " ... (we) should rejoice at the emergence of a new subject ... rich and deep, with all the theorems yet to be proved." His remarks are as true today as they were then. ix

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