Evolutionary biology is an essential basic science for medicine, but few doctors and medical researchers are familiar with its most relevant principles. Most medical schools have geneticists who understand evolution, but few have even one evolutionary biologist to suggest other possible applications. The canyon between evolutionary biology and medicine is wide. The question is whether they offer each other enough to make bridge building worthwhile. What benefits could be expected if evolution were brought fully to bear on the problems of medicine? How would studying medical problems advance evolutionary research? Do doctors need to learn evolution, or is it valuable mainly for researchers? What practical steps will promote the application of evolutionary biology in the areas of medicine where it offers the most? To address these questions, we review current and potential applications of evolutionary biology to medicine and public health. Some evolutionary technologies, such as population genetics, serial transfer production of live vaccines, and phylogenetic analysis, have been widely applied. Other areas, such as infectious disease and aging research, illustrate the dramatic recent progress made possible by evolutionary insights. In still other areas, such as epidemiology, psychiatry, and understanding the regulation of bodily defenses, applying evolutionary principles remains an open opportunity. In addition to the utility of specific applications, an evolutionary perspective fundamentally challenges the prevalent but fundamentally incorrect metaphor of the body as a machine designed by an engineer. Bodies are vulnerable to disease - and remarkably resilient - precisely because they are not machines built from a plan. They are, instead, bundles of compromises shaped by natural selection in small increments to maximize reproduction, not health. Understanding the body as a product of natural selection, not design, offers new research questions and a framework for making medical education more coherent. We conclude with recommendations for actions that would better connect evolutionary biology and medicine in ways that will benefit public health. It is our hope that faculty and students will send this article to their undergraduate and medical school Deans, and that this will initiate discussions about the gap, the great opportunity, and action plans to bring the full power of evolutionary biology to bear on human health problems.