Most solid human tumours are aneuploid, that is, they contain an abnormal number of chromosomes. Paradoxically, however, aneuploidy has been reported to induce a stress response that suppresses cellular proliferation in vitro. Here, we review the progress in our understanding of the causes and effects of aneuploidy in cancer and discuss how, in specific contexts, aneuploidy can provide a growth advantage and facilitate cellular transformation. We also explore the emerging possibilities for targeting the cause or consequences of aneuploidy therapeutically.