Genetic changes arising in human pluripotent stem cells (hPSC) upon culture may bestow unwanted or detrimental phenotypes to cells, thus potentially impacting on the applications of hPSCs for clinical use and basic research. In the 20 years since the first report of culture‐acquired genetic aberrations in hPSCs, a characteristic spectrum of recurrent aberrations has emerged. The preponderance of such aberrations implies that they provide a selective growth advantage to hPSCs upon expansion. However, understanding the consequences of culture‐acquired variants for specific applications in cell therapy or research has been more elusive. The rapid progress of hPSC‐based therapies to clinics is galvanizing the field to address this uncertainty and provide definitive ways both for risk assessment of variants and reducing their prevalence in culture. Here, we aim to provide a timely update on almost 20 years of research on this fascinating, but a still unresolved and concerning, phenomenon.