We consider the propagation of classical and quantum strings on cosmological spacetimes which interpolate from a collapsing phase to an expanding phase. We begin by considering the classical propagation of strings on spacetimes 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, while 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 than
√
α
′
inevitably emerge in highly excited states, and that in this regime there is an interesting correspondence between strings on anisotropic cosmological spacetimes 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.