Both embryonic and adult stem cells are endowed with a superior capacity to prevent the accumulation of genetic lesions, repair them, or avoid their propagation to daughter cells, which would be particularly detrimental to the whole organism. Inducible pluripotent stem cells also display a robust DNA damage response, but the stability of their genome is often conditioned by the mutational history of the cell population of origin, which constitutes an obstacle to clinical applications. Cancer stem cells are particularly tolerant to DNA damage and fail to undergo senescence or regulated cell death upon accumulation of genetic lesions. Such a resistance contributes to the genetic drift of evolving tumors as well as to their limited sensitivity to chemo- and radiotherapy. Here, we discuss the pathophysiological and therapeutic implications of the molecular pathways through which stem cells cope with DNA damage.