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Sudden stratospheric warmings (SSWs) are impressive fluid dynamical events in which large and rapid temperature increases in the winter polar stratosphere (~10 to 50km) are associated with a complete reversal of the climatological wintertime westerly winds. SSWs are caused by the breaking of planetary-scale waves that propagate upwards from the troposphere. During an SSW, the polar vortex breaks down, accompanied by rapid descent and warming of air in polar latitudes, mirrored by ascent and cooling above the warming. The rapid warming and descent of the polar air column affects tropospheric weather, shifting jet streams, storm tracks, and the Northern Annular Mode, making cold air outbreaks over North America and Eurasia more likely. SSWs affect the atmosphere above the stratosphere, producing widespread effects on atmospheric chemistry, temperatures, winds, neutral (non-ionized) particles and electron densities, and electric fields. These effects span both hemispheres. Given their crucial role in the whole atmosphere, SSWs are also seen as a key process to analyze in climate change studies and subseasonal to seasonal prediction. This work reviews the current knowledge on the most important aspects of SSWs, from the historical background to dynamical processes, modelling, chemistry, and impact on other atmospheric layers.