Many human diseases are the result of evolutionary processes on time scales much shorter than the human lifetime. Prominent examples of pathogenic, measurably evolving populations are cancer cells in a tumor and infectious parasites, such as bacteria and viruses. Cancer progression is driven by mutation and selection in an asexually reproducing population of tumor cells. Treatment of these constantly changing ensembles of genetically heterogeneous cells is complicated by evolutionary escape from the selective pressure of drugs and immune responses. We present mathematical models for the evolutionary dynamics of escape and discuss applications to the genetic progression of cancer. A graphical model for the accumulation of genetic changes is shown to improve genetics-based survival predictions in patients with renal cell carcinoma. We also discuss statistical methods for inferring the genetic diversity of tumors from ultra-deep sequencing experiments and compare experimental results to model predictions.