El Niño and La Niña events in the tropical Pacific have significant and disrupting impacts on the global atmospheric and oceanic circulation. El Niño Southern Oscillation (ENSO) impacts also extend above the troposphere, affecting the strength and variability of the stratospheric polar vortex in the high latitudes of both hemispheres, as well as the composition and circulation of the tropical stratosphere. El Niño events are associated with a warming and weakening of the polar vortex in the polar stratosphere of both hemispheres, while a cooling can be observed in the tropical lower stratosphere. These impacts are linked by a strengthened Brewer-Dobson circulation. Anomalous upward wave propagation is observed in the extratropics of both hemispheres. For La Niña, these anomalies are often opposite. The stratosphere in turn affects surface weather and climate over large areas of the globe. Since these surface impacts are long-lived, the changes in the stratosphere can lead to improved surface predictions on time scales of weeks to months. Over the past decade, our understanding of the mechanisms through which ENSO can drive impacts remote from the tropical Pacific has improved. This study reviews the possible mechanisms connecting ENSO to the stratosphere in the tropics and the extratropics of both hemispheres while also considering open questions, including nonlinearities in the teleconnections, the role of ENSO diversity, and the impacts of climate change and variability.