The polarisation properties of light are often used as a means of information transfer, as well as in assessing the structural and compositional properties of materials. In many practical situations, however, the propagation of polarised light is impeded by random scattering, which tends to scramble the information content of the underlying fields. Theoretical study of the random scattering of light is an old, complex field of research, in which approximate, numerical methods are often favoured over exact mathematical analysis. Random matrix theory, in which systems are modelled using random scattering matrices, has uncovered universal properties of wide classes of random media, most notably in the field of quantum scattering. Despite also finding success in optics, a random matrix theory of polarised light has yet to be pursued.

In this thesis we apply the notion of random matrices to develop statistical techniques for modelling the random scattering of polarised light. We present a full derivation of the symmetries of the vectorial scattering and transfer matrices that describe dielectric scattering media, including the scattering of evanescent wave components. We then consider the circular ensembles as a simple random matrix model and explore its implications for the scattering of polarised light. Moving beyond elementary models, a rigorous, statistical theory of the scattering matrix for discrete random media is presented, and exact mathematical results are derived in certain special cases. A numerical simulation method for studying scattering matrices describing random media of arbitrary thickness is then developed and validated against known physical phenomena. Finally, the techniques developed within this thesis are applied to the problem of the recovery of polarisation information within light that has propagated through a random medium.