It is widely accepted that magnetic reconnection is the most efficient means of transporting solar wind plasma across the terrestrial magnetopause. However, at Saturn's magnetopause, conditions for reconnection are considerably different to those at the Earth, and reconnection seems to play a much less important role in energy transport across the magnetopause at Saturn than at the Earth. Therefore, other transport mechanisms, such as diffusion and 'viscous' mechanisms become relatively more important. The Kelvin-Helmholtz (K-H) instability is a prime candidate for such an alternative mechanism, as first suggested by Dungey (1955). The K-H instability has been the subject of numerous studies at the Earth, but its operation and impact at Saturn is much less well understood. This thesis builds on work undertaken by previous authors to identify K-H waves on the Kronian magnetopause. These waves were found to exist on the dusk flank magnetopause, which has previously been thought to be stable to the instability due to a reduced flow shear between magnetospheric and magnetosheath plasma compared to the dawn flank. This asymmetry is due to the sense of magnetospheric (sub)corotation with the planet. The frequency with which the magnetopause is seen to become K-H unstable as a function of local time is then investigated, and surprisingly, the dusk flank is found to be as unstable as the dawn flank. It is proposed that this is due to the nature of the magnetic field structure in the outer magnetosphere adjacent to the magnetopause. Viscous interaction causes the magnetic field in the outer magnetosphere to have a larger component parallel to the magnetopause in the equatorial plane on the dawn flank, compared to the dusk flank. This in turn raises the threshold for boundary instability on the dawn flank magnetopause. The final part of the thesis investigates the non-linear behaviour of the instability far along the flanks- more specifically concerning the rolling up of waves into vortices and the plasma mixing that results.