'One of the great legacies of the classification of the finite simple groups is the existence of the Monster $ldots$. Work of Borcherds and Frenkel-Lepowsky-Meurman led to the notion of a vertex (operator) algebra, which was seen to be the same as the chiral algebras used by physicists in conformal field theory $ldots$. 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' - from the Preface.This volume contains the proceedings of a Joint Summer Research Conference held at Mount Holyoke College in June 1994. As perhaps the first conference proceedings devoted exclusively to the subject known as 'Moonshine', this work contains something for many mathematicians and physicists. It features: results concerning the monster simple group and other simple groups; connections with elliptic cohomology; connections with 2-dimensional conformal field theory; the role of operads; and, connections with modular functions. ""Much of Moonshine, the Monster, and Related Topics"" features new results not available anywhere else.