  We study how the spin-statistics theorem relates to the geometric structures
on phase space that are introduced in quantisation procedures (namely a U(1)
bundle and connection). The relation can be proved in both the relativistic and
the non-relativistic domain (in fact for any symmetry group including internal
symmetries) without quantum field theory, by the requirement that the exchange
can be implemented smoothly by a class of symmetry transformations that project
in the phase space of the joint system. We discuss the interpretation of this
requirement, stressing the fact that any distinction of identical particles
comes solely from the choice of coordinates. We then examine our construction
in the geometric and the coherent-state-path-integral quantisation schemes. In
the appendix we apply our results to exotic systems exhibiting continuous
``spin'' and ``fractional statistics''. This gives novel and unusual forms of
the spin-statistics relation.
