Human pluripotent stem cells (hPSC) are known to acquire chromosomal abnormalities, which range from point mutations to large copy number changes, including full chromosome aneuploidy. These aberrations have a wide-ranging influence on the state of cells, in both the undifferentiated and differentiated state. Currently, very little is known on how these abnormalities will impact the clinical translation of hPSC, and particularly their potential to prime cells for oncogenic transformation. A further complication is that many of these abnormalities exist in a mosaic state in culture, which complicates their detection with conventional karyotyping methods. In this review we discuss current knowledge on how these aberrations influence the cell state and how this may impact the future of research and the cells’ clinical potential.