Owing to a unique set of attributes, human pluripotent stem cells (hPSCs) have emerged as a promising cell source for regenerative medicine, disease modeling and drug discovery. Assurance of genetic stability over long term maintenance of hPSCs is pivotal in this endeavor, but hPSCs can adapt to life in culture by acquiring non-random genetic changes that render them more robust and easier to grow. In separate studies between 12.5% and 34% of hPSC lines were found to acquire chromosome abnormalities over time, with the incidence increasing with passage number. The predominant genetic changes found in hPSC lines involve changes in chromosome number and structure (particularly of chromosomes 1, 12, 17 and 20), reminiscent of the changes observed in cancer cells. In this review, we summarize current knowledge on the causes and consequences of aneuploidy in hPSCs and highlight the potential links with genetic changes observed in human cancers and early embryos. We point to the need for comprehensive characterization of mechanisms underpinning both the acquisition of chromosomal abnormalities and selection pressures, which allow mutations to persist in hPSC cultures. Elucidation of these mechanisms will help to design culture conditions that minimize the appearance of aneuploid hPSCs. Moreover, aneuploidy in hPSCs may provide a unique platform to analyse the driving forces behind the genome evolution that may eventually lead to cancerous transformation.