  We consider the propagation of classical and quantum strings on cosmological
space-times which interpolate from a collapsing phase to an expanding phase. We
begin by considering the classical propagation of strings on space-times with
isotropic and anisotropic cosmological singularities. We find that cosmological
singularities fall into two classes, in the first class the string evolution is
well behaved all the way up to the singularity, whilst in the second class it
becomes ill-defined. Then assuming the singularities are regulated by string
scale corrections, we consider the implications of the propagation through a
`bounce'. It is known that as we evolve through a bounce, quantum strings will
become excited giving rise to `particle transmutation'. We reconsider this
effect, giving qualitative arguments for the amount of excitation for each
class. We find that strings whose physical wavelength at the bounce is less
that $\sqrt{\alpha'}$ inevitably emerge in highly excited states, and that in
this regime there is an interesting correspondence between strings on
anisotropic cosmological space-times and plane waves. We argue that long
wavelength modes, such as those describing cosmological perturbations, will
also emerge in mildly excited string scale mass states. Finally we discuss the
relevance of this to the propagation of cosmological perturbations in models
such as the ekpyrotic/cyclic universe.
