Viruses, bacteria, eukaryotic parasites, cancer cells, agricultural pests and other inconvenient animates have an unfortunate tendency to escape from selection pressures that are meant to control them. Chemotherapy, anti-viral drugs or antibiotics fail because their targets do not hold still, but evolve resistance. A major problem in developing vaccines is that microbes evolve and escape from immune responses. The fundamental question is the following: if a genetically diverse population of replicating organisms is challenged with a selection pressure that has the potential to eradicate it, what is the probability that this population will produce escape mutants? Here, we use multi-type branching processes to describe the accumulation of mutants in independent lineages. We calculate escape dynamics for arbitrary mutation networks and fitness landscapes. Our theory shows how to estimate the probability of success or failure of biomedical intervention, such as drug treatment and vaccination, against rapidly evolving organisms.