  The current status of `Mathieu Moonshine', the idea that the Mathieu group
M24 organises the elliptic genus of K3, is reviewed. While there is a
consistent decomposition of all Fourier coefficients of the elliptic genus in
terms of Mathieu M24 representations, a conceptual understanding of this
phenomenon in terms of K3 sigma-models is still missing. In particular, it
follows from the recent classification of the automorphism groups of arbitrary
K3 sigma-models that (i) there is no single K3 sigma-model that has M24 as an
automorphism group; and (ii) there exist `exceptional' K3 sigma-models whose
automorphism group is not even a subgroup of M24. Here we show that all cyclic
torus orbifolds are exceptional in this sense, and that almost all of the
exceptional cases are realised as cyclic torus orbifolds. We also provide an
explicit construction of a Z5 torus orbifold that realises one exceptional
class of K3 sigma-models.
