Understanding how impacts may differ across alternative levels of future climate change is necessary to inform mitigation and adaptation measures. The Benefits of Reduced Anthropogenic Climate changE (BRACE) project assesses the differences in impacts between two specific climate futures: a higher emissions future with global average temperature increasing about 3.7 A degrees C above pre-industrial levels toward the end of the century and a moderate emissions future with global average warming of about 2.5 A degrees C. BRACE studies in this special issue quantify avoided impacts on physical, managed, and societal systems in terms of extreme events, health, agriculture, and tropical cyclones. Here we describe the conceptual framework and design of BRACE and synthesize its results. Methodologically, the project combines climate modeling, statistical analysis, and impact assessment and draws heavily on large ensembles using the Community Earth System Model. It addresses uncertainty in future societal change by employing two pathways for future socioeconomic development. Results show that the benefits of reduced climate change within this framework vary substantially across types of impacts. In many cases, especially related to extreme heat events, there are substantial benefits to mitigation. The benefits for some heat extremes are statistically significant in some regions as early as the 2020s and are widespread by mid-century. Benefits are more modest for agriculture and exposure to some health risks. Benefits are negative for agriculture when CO2 fertilization is incorporated. For several societal impacts, the effect on outcomes of alternative future societal development pathways is substantially larger than the effect of the two climate scenarios.