  In this talk we consider the relationship between the conjectured uniqueness
of the Moonshine module of Frenkel, Lepowsky and Meurman and Monstrous
Moonshine, the genus zero property for Thompson series discovered by Conway and
Norton. We discuss some evidence to support the uniqueness of the Moonshine
module by considering possible alternative orbifold constructions from a Leech
lattice compactified string. Within these constructions we find a new
relationship between the centralisers of the Monster group and the Conway group
generalising an observation made by Conway and Norton. We also relate the
uniqueness of the Moonshine module to Monstrous Moonshine and argue that given
this uniqueness, then the genus zero properties hold if and only if orbifolding
the Moonshine module with respect to a Monster element reproduces the Moonshine
module or the Leech theory. (Talk presented at the Nato Advanced Research
Workshop on `Low dimensional topology and quantum field theory`, Cambridge,
6-13 Sept 1992)
