In this talk I will survey the history of Moonshine.
In 1978 John McKay remarked on a curiosity about the Fourier coefficients of one particular function: The j-invariant. They seemed to be related to the representation theory of the Monster group. At the time he would have hardly guessed that his observation would initiate a whole new research area in mathematics and eventually earn Richard Borcherds a Fields Medal.
McKay had found what nowadays is known to be the character of a vertex operator algebra with exceptionally large symmetry group, the Monster VOA. Following Borcherds proof of the foundational Conway-Norton conjecture, published in 1992, VOA theory consolidated. The character theory of more general VOAs was examined and finally settled in 1996 by Zhu. About ten years later, in 2005, Huang recast this theory using modular tensor categories.
As in 1978, computations by Eguchi-Ooguri-Tachikawa from 2010 have shaken up the field: The elliptic genus of K3 surfaces seems to be related to the Mathieu group. There still is not satisfactory explanation in sight. Instead, more and more groups are related to various Moonshine phenomena. This might not yield another Fields Medal, in any case exciting mathematics is about to emerge.