The next generation of nuclear reactors will expose materials to conditions that, in some cases, are even more extreme than those in current fission reactors, inevitably leading to new materials science challenges. Radiation-induced damage and corrosion are two key phenomena that must be understood both independently and synergistically, but their interactions are often convoluted. In the light water reactor community, a tremendous amount of work has been done to illuminate irradiation-corrosion effects, and similar efforts are under way for heavy liquid metal and molten salt environments. While certain effects, such as radiolysis and irradiation-assisted stress corrosion cracking, are reasonably well established, the basic science of how irradiation-induced defects in the base material and the corrosion layer influence the corrosion process still presents many unanswered questions. In this review, we summarize the work that has been done to understand these coupled extremes, highlight the complex nature of this problem, and identify key knowledge gaps. Expected final online publication date for the Annual Review of Materials Science, Volume 51 is August 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.