Limited progress with mitigation makes it almost inevitable that global warming of 1.5°C will be exceeded. This realization confronts Parties to the United Nations Framework Convention on Climate Change (UNFCCC) with a choice either to stabilize warming above but as close as possible to 1.5°C or to reverse global warming back to this level. We review core concepts and current knowledge relating to overshoot: an exceedance and subsequent decline back below a specified global warming level. We clarify the concept and origins of overshoot in science and climate policy, discuss the key drivers of climate-related risks and how they might evolve under overshoot trajectories to foster more systematic research into those risks, and consider the role of adaptation. We then consider the feasibility of overshoot in terms of mitigation across the six feasibility dimensions introduced by the Intergovernmental Panel on Climate Change (IPCC) in its sixth Assessment Report. We conclude by discussing critical barriers, challenges, and knowledge gaps related to overshoot.