Among the great diversity of atmospheric circulation patterns observed throughout the solar system, polar vortices stand out as a nearly ubiquitous planetary-scale phenomenon. In recent years there have been significant advances in the observation of planetary polar vortices, culminating in the fascinating discovery of Jupiter's polar vortex clusters during the Juno mission. Alongside these observational advances has been a major effort to understand polar vortex dynamics using theory, idealised and comprehensive numerical models, and laboratory experiments. Here we review our current knowledge of planetary polar vortices, highlighting both the diversity of their structures, as well as fundamental dynamical similarities. We propose a new convention of vortex classification, which adequately captures all those observed in our solar system, and demonstrates the key role of polar vortices in the global circulation, transport, and climate of all planets. We discuss where knowledge gaps remain, and the observational, experimental, and theoretical advances needed to address them. In particular, as the diversity of both solar system and exoplanetary data increases exponentially, there is now a unique opportunity to unify our understanding of polar vortices under a single dynamical framework.