Microbial drug persistence is a widespread phenomenon in which a subpopulation of microorganisms is able to survive antimicrobial treatment without acquiring resistance-conferring genetic changes. Microbial persisters can cause recurrent or intractable infections, and, like resistant mutants, they carry an increasing clinical burden. In contrast to heritable drug resistance, however, the biology of persistence is only beginning to be unraveled. Persisters have traditionally been thought of as metabolically dormant, nondividing cells. As discussed in this review, increasing evidence suggests that persistence is in fact an actively maintained state, triggered and enabled by a network of intracellular stress responses that can accelerate processes of adaptive evolution. Beyond shedding light on the basis of persistence, these findings raise the possibility that persisters behave as an evolutionary reservoir from which resistant organisms can emerge. As persistence and its consequences come into clearer focus, so too does the need for clinically useful persister-eradication strategies.