Aneuploidy, an irregular number of chromosomes in cells, is a hallmark feature of cancer. Aneuploidy results from chromosomal instability (CIN) and occurs in almost 90% of all tumours. While many cancers display an ongoing CIN phenotype, cells can also be aneuploid without displaying CIN. CIN drives tumour evolution as ongoing chromosomal missegregation will yield a progeny of cells with variable aneuploid karyotypes. The resulting aneuploidy is initially toxic to cells because it leads to proteotoxic and metabolic stress, cell cycle arrest, cell death, immune cell activation and further genomic instability. In order to overcome these aneuploidy-imposed stresses and adopt a malignant fate, aneuploid cancer cells must develop aneuploidy-tolerating mechanisms to cope with CIN. Aneuploidy-coping mechanisms can thus be considered as promising therapeutic targets. However, before such therapies can make it into the clinic, we first need to better understand the molecular mechanisms that are activated upon aneuploidization and the coping mechanisms that are selected for in aneuploid cancer cells. In this review, we discuss the key biological responses to aneuploidization, some of the recently uncovered aneuploidy-coping mechanisms and some strategies to exploit these in cancer therapy.