Research during the last two decades has established that variability of the winter polar stratospheric vortex can significantly influence the troposphere, affecting the likelihood of extreme weather events and the skill of long-range weather forecasts. This influence is particularly strong following the rapid breakdown of the vortex in events known as sudden stratospheric warmings (SSWs). This thesis addresses some outstanding issues in our understanding of the dynamics of this stratospheric variability and its influence on the troposphere. First, a geometrical method is developed to characterise two-dimensional polar vortex variability. This method is also able to identify types of SSW in which the vortex is displaced from the pole and those in which it is split in two; known as displaced and split vortex events. It shown to capture vortex variability at least as well as previous methods, but has the advantage of being easily applicable to climate model simulations. This method is subsequently applied to 13 stratosphere-resolving climate models. Almost all models show split vortex events as barotropic and displaced vortex events as baroclinic; a difference also seen in observational reanalysis data. This supports the idea that split vortex events are caused by a resonant excitation of the barotropic mode. Models show consistent differences in the surface response to split and displaced vortex events which do not project stongly onto the annular mode. However, these differences are approximately co-located with lower stratospheric anomalies, suggesting that a local adjustment to stratospheric potential vorticity anomalies is the mechanism behind the different surface responses. Finally, the predictability of the polar stratosphere and its influence on the troposphere is assessed in a stratosphere-resolving seasonal forecast system. Little skill is found in the prediction of the strength of the Northern Hemisphere vortex at lead times beyond one month. However, much greater skill is found for the Southern Hemisphere vortex during austral spring. This allows for forecasts of interannual changes in ozone depletion to be inferred at lead times much beyond previous forecasts. It is further demonstrated that this stratospheric skill descends with time and leads to an enhanced surface skill at lead times of more than three months.