We urgently need to predict species responses to climate change to minimize future biodiversity loss and ensure we do not waste limited resources on ineffective conservation strategies. Currently, most predictions of species responses to climate change ignore the potential for evolution. However, evolution can alter species ecological responses, and different aspects of evolution and ecology can interact to produce complex eco-evolutionary dynamics under climate change. Here we review how evolution could alter ecological responses to climate change on species warm and cool range margins, where evolution could be especially important. We discuss different aspects of evolution in isolation, and then synthesize results to consider how multiple evolutionary processes might interact and affect conservation strategies. On species cool range margins, the evolution of dispersal could increase range expansion rates and allow species to adapt to novel conditions in their new range. However, low genetic variation and genetic drift in small range-front populations could also slow or halt range expansions. Together, these eco-evolutionary effects could cause a three-step, stop-and-go expansion pattern for many species. On warm range margins, isolation among populations could maintain high genetic variation that facilitates evolution to novel climates and allows species to persist longer than expected without evolution. This 'evolutionary extinction debt' could then prevent other species from shifting their ranges. However, as climate change increases isolation among populations, increasing dispersal mortality could select for decreased dispersal and cause rapid range contractions. Some of these eco-evolutionary dynamics could explain why many species are not responding to climate change as predicted. We conclude by suggesting that resurveying historical studies that measured trait frequencies, the strength of selection, or heritabilities could be an efficient way to increase our eco-evolutionary knowledge in climate change biology.