  Eguchi, Ooguri and Tachikawa have observed that the elliptic genus of type II
string theory on K3 surfaces appears to possess a Moonshine for the largest
Mathieu group. Subsequent work by several people established a candidate for
the elliptic genus twisted by each element of M24. In this paper we prove that
the resulting sequence of class functions are true characters of M24, proving
the Eguchi-Ooguri-Tachikawa conjecture. We prove the evenness property of the
multiplicities, as conjectured by several authors. We also identify the role
group cohomology plays in both K3-Mathieu Moonshine and Monstrous Moonshine; in
particular this gives a cohomological interpretation for the non-Fricke
elements in Norton's Generalised Monstrous Moonshine conjecture. We investigate
the intriguing proposal of Gaberdiel-Hohenegger-Volpato that K3-Mathieu
Moonshine lifts to the Conway group Co1.
