A new modular, free software package is described for tracking tropopause polar vortices (TPVs) natively on structured or unstructured grids. Motivated by limitations in spatial characterization and time tracking within existing approaches, TPVTrack leverages the dynamics of TPVs to represent their (1) spatial structure, with variable shapes and intensities and (2) time evolution, with mergers and splits. TPVs are segmented from the gridded flow field into spatial objects as restricted regional watershed basins on the tropopause, described by geometric metrics, associated over time by overlap similarity into major and minor correspondences, and tracked along major correspondences. Simplified segmentation and correspondence test cases illustrate some of the appeal, sensitivities, and limitations of TPVTrack, including effective representation of spatial shape and reduced false positive associations in time. Tracked TPVs in more realistic historical conditions are consistent in bulk with expectations of life cycle and mean structure. Individual tracks are less reliable when discriminating between multiple overlaps. Modifications to track other physical features are possible, with each application requiring evaluation.