The Monster group is a quite famous and strange creature which appears in the classification of simple finite groups. In the 70's, the mathematician John McKay noticed a surprising link between the representation theory of the Monster and an important modular function, called the J-invariant. This remark was first underestimated and considered as a meaningless coincidence. However, first Thompson, and then Conway and Norton found more evidences for such a link and formulated the Moonshine conjectures. They were totally proved by Borcherds in 1992 by using the new formalism of vertex algebras, and he received the Field medal in 1998 in part for this work.
In this talk, we would like to introduce the actors (mathematicians and mathematical objects) involved in the story of Moonshine conjectures and relate this story. We will define the Monster group and the J-invariant, and state the Moonshine conjectures and their link with vertex algebras. We will also give the principal steps in the history of the resolution of the conjectures.