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Sudden stratospheric warmings (SSWs) are a key driver of winter climate variability in the Northern Hemisphere. SSWs are a disruption of the strong stratospheric westerlies over the winter pole in which the winds in the upper to middle stratosphere, from about 30 to 50 km above the surface, weaken and reverse and the polar cap temperatures increase by up to 50 K in only a few days. These events affect tropospheric conditions for the two months following, on average shifting the North Atlantic storm track equatorward and resulting in a negative Northern Annular Mode and North Atlantic Oscillation at the surface. These changes are associated with colder and drier than average conditions in Northern Europe and Eurasia and warmer and wetter than average conditions across Southern Europe, as well as high temperatures across North Africa, the Middle East, and Central Asia and increased cold air outbreaks in North America and Eurasia. This thesis examines this typical surface response to SSWs in several different contexts. We consider its relationship to other atmospheric phenomena and features, first quantifying its importance relative to the North Atlantic impacts of the El Niño-Southern Oscillation (ENSO) and then examining the role of ozone chemistry in modeling the surface response to SSWs. We also study the variability of the surface signature of SSWs, with the goal of understanding the uncertainty in magnitude and spatial pattern of surface climate patterns following SSWs and the relative roles of different sources of this uncertainty. After providing background and context in the first chapter, the second chapter studies interactions between SSWs and the El Niño phase of ENSO. El Niño affects climate in the North Atlantic and European regions, those most affected by SSWs, through tropospheric and stratospheric pathways. One of these pathways is increased SSW frequency. However, most SSWs (about 90\%) are unrelated to ENSO, and the importance for boreal winter surface climate of this frequency increase compared to other El Niño pathways remains to be quantified. We here contrast these two sources of variability using two 200-member ensembles of one-year integrations of the Whole Atmosphere Community Climate Model, one ensemble with prescribed El Niño sea surface temperatures (SSTs) and one with neutral-ENSO SSTs. We form composites of wintertime climate anomalies, with and without SSWs, in each ensemble and contrast them to a basic state represented by neutral-ENSO winters without SSWs. This approach allows us to isolate the distinct effects of ENSO and SSWs more clearly than was done in previous work. We find that El Niño and SSWs both result in negative North Atlantic Oscillation anomalies and have comparable impacts on European precipitation, but SSWs cause larger Eurasian cooling. These results indicate the potential impact of a strong El Niño on seasonal forecasting in the North Atlantic as well as the importance of resolving the stratosphere in subseasonal and seasonal forecast models to best capture stratospheric polar vortex variability. In the third chapter, we study the importance of interactive ozone chemistry in representing the stratospheric polar vortex and Northern Hemisphere winter surface climate variability. Modeling and observational studies have reported effects of stratospheric ozone extremes on Northern Hemisphere spring climate. Recent work has further suggested that the coupling of ozone chemistry and dynamics amplifies the surface response to midwinter SSWs. We contrast two 200-year simulations from the interactive and specified chemistry (and thus ozone) versions of the Whole Atmosphere Community Climate Model with constant year-2000 forcings. This experiment is thus designed to clearly isolate the impact of interactive ozone on polar vortex variability. In particular, we analyze the response with and without interactive chemistry to midwinter SSWs, March SSWs, and strong polar vortex events (SPVs). With interactive chemistry, the stratospheric polar vortex is stronger, and more SPVs occur, but we find little effect on the frequency of midwinter SSWs. At the surface, interactive chemistry results in a pattern resembling a more negative North Atlantic Oscillation following midwinter SSWs, but with little impact on the surface signatures of late winter SSWs and SPVs. These results suggest that including interactive ozone chemistry in model simulations is important for representing North Atlantic and European winter climate variability. In the fourth chapter, we turn from models to reanalysis and consider the uncertainty in the surface response to SSWs. While the qualitative features of the mean surface signature of SSWs in the North Atlantic and Europe are well-established, its uncertainties as well as other features of surface climate following SSWs are less well-understood. To address the question of robustness of the mean observed response to SSWs, we use bootstrapping with replacement to construct synthetic SSW composites from SSW events in reanalysis, creating an ensemble of composites comparable to the observed one. We then examine the differences across these synthetic composites. We find that the canonical responses of a negative North Atlantic Oscillation and associated temperature and precipitation anomalies in the North Atlantic and European regions in the months following SSWs are robust. However, the magnitude and spatial pattern of these anomalies vary considerably across the composites. We further find that this uncertainty is unrelated to vortex strength and is instead the result of unrelated tropospheric variability. These results have implications for evaluating the fidelity of forecast models in capturing the surface impact of SSWs, by comparing both the mean impact as well as the contribution from internal variability with observations. Overall, we demonstrate the complexity of interactions of sudden stratospheric warmings with other sources of variability in the Earth system. We find that the state of the polar vortex itself, the strength of downward propagation following the SSW, and the surface response can all be affected in important ways by these other components (e.g. tropospheric variability and Arctic ozone). We close by providing broader context for these results and looking towards continuing and future work in the field.