Aftershocks can compound the impacts of a major earthquake, disrupting recovery efforts and potentially further damaging weakened buildings and infrastructure. Forecasts of the probability of aftershocks can therefore aid decision-making during earthquake response and recovery. Several countries issue authoritative aftershock forecasts. Most aftershock forecasts are based on simple statistical models that were first developed in the 1980s and remain the best available models. We review these statistical models, and the wide-ranging research to advance aftershock forecasting through better statistical, physical, and machine learning methods. Physics-based forecasts based on mainshock stress changes can sometimes match the statistical models in testing, but don’t yet outperform them. Physical models are also hampered by unsolved problems such as the mechanics of dynamic triggering and the influence of background conditions. Initial work on machine learning forecasts shows promise, and new machine learning earthquake catalogs provide an opportunity to advance all types of aftershock forecasts.