Evaluation of the extent and nature of induced pluripotent stem cell (iPSC) genetic instability is important for both basic research and future clinical use. As previously demonstrated regarding embryonic stem cells, such DNA aberrations might affect the differentiation capacity of the cells and increase their tumorigenicity. Here, we first focus on the contribution of multiple DNA damage response pathways during cellular reprogramming. We then discuss the origin and mechanisms responsible for the modification of genetic material in iPSCs (pre-existing variations in somatic cells, mutations induced by reprogramming factors, and mutations induced by culture expansion) and deepen the possible functional consequences of genetic variations in these cells. Lastly, we present some recent improvements of iPSC generation methods aimed at obtaining cells with fewer genetic variations.