Our knowledge of the dynamics and masses of galaxies in the Local Group has long been limited by the fact that only line-of-sight velocities were observationally accessible. This introduces significant degeneracies in dynamical models, which can only be resolved by measuring also the velocity components perpendicular to the line of sight. However, beyond the solar neighborhood, the corresponding proper motions have generally been too small to measure. This has changed dramatically over the past decade, especially due to the angular resolution and stability available on the Hubble Space Telescope. Proper motions can now be reliably measured throughout the Local Group, as illustrated by, e.g., the work of the HSTPROMO collaboration. In this review, I summarize the importance of proper motions for Local Group science, and I describe the current and future observational approaches and facilities available to measure proper motions. I highlight recent results on various Milky Way populations (globular clusters, the bulge, the metal-poor halo, hypervelocity stars, and tidal streams), dwarf satellite galaxies, the Magellanic Clouds and the Andromeda System.