Although we can treat cancers with cytotoxic chemotherapies, target them with molecules that inhibit oncogenic drivers, and induce substantial cell death with radiation, local and metastatic tumours recur, resulting in extensive morbidity and mortality. Indeed, driving a tumour to extinction is difficult. Geographically dispersed species of organisms are perhaps equally resistant to extinction, but >99.9% of species that have ever existed on this planet have become extinct. By contrast, we are nowhere near that level of success in cancer therapy. The phenomena are broadly analogous-in both cases, a genetically diverse population mutates and evolves through natural selection. The goal of cancer therapy is to cause cancer cell population extinction, or at least to limit any further increase in population size, to prevent the tumour burden from overwhelming the patient. However, despite available treatments, complete responses are rare, and partial responses are limited in duration. Many patients eventually relapse with tumours that evolve from cells that survive therapy. Similarly, species are remarkably resilient to environmental change. Paleontology can show us the conditions that lead to extinction and the characteristics of species that make them resistant to extinction. These lessons could be translated to improve cancer therapy and prognosis.