Mutations that drive uncontrolled cell-cycle progression are requisite events in tumorigenesis. But evolution has installed in the proliferative programmes of mammalian cells a variety of innate tumour-suppressive mechanisms that trigger apoptosis or senescence, should proliferation become aberrant. These contingent processes rely on a series of sensors and transducers that act in a coordinated network to target the machinery responsible for apoptosis and cell-cycle arrest at different points. Although oncogenic mutations that disable such networks can have profound and varied effects on tumour evolution, they may leave intact latent tumour-suppressive potential that can be harnessed therapeutically.