In late 1970's John McKay discovered the astonishing identity 196884=196883+1, which lead Conway and Norton to formulate the famous Monstrous Moonshine conjectures about the Monster group, the largest sporadic finite simple group. The simplest part of the conjecture is about the existence of a natural infinite dimensional Z-graded representation of the Monster, with the dimensions of the graded pieces coinciding with the coefficients of the j-function, which is one of the most fundamental objects in number theory. Such a module was constructed by I.Frenkel-Lepowsky-Meurman in mid 80's with an extra algebraic structure - a vertex operator algebra - whose full automorphism group is proved to be the Monster. Later, most of these conjectures were proved by Borcherds.
In this talk, I will give an elementary historical introduction to formulation of these conjectures and some of the subsequent mathematical developments. Throughout the talk, some knowledge on basic complex analysis is assumed.