  Generalised moonshine is reviewed from the point of view of holomorphic
orbifolds, putting special emphasis on the role of the third cohomology group
H^3(G, U(1)) in characterising consistent constructions. These ideas are then
applied to the case of Mathieu moonshine, i.e. the recently discovered
connection between the largest Mathieu group M_24 and the elliptic genus of K3.
In particular, we find a complete list of twisted twining genera whose modular
properties are controlled by a class in H^3(M_24, U(1)), as expected from
general orbifold considerations.
