When a human cell detects damaged DNA, it initiates the DNA damage response (DDR) that permits it to repair the damage and avoid transmitting it to daughter cells. Despite this response, changes to the genome occur and some cells, such as proliferating cancer cells, are prone to genome instability. The cellular processes that lead to genomic changes after a genotoxic event are not well understood. Our research focuses on the relationship between genotoxic cancer drugs and checkpoint adaptation, which is the process of mitosis with damaged DNA. We examine the types of DNA damage induced by widely used cancer drugs and describe their effects upon proliferating cancer cells. There is evidence that cell death caused by genotoxic cancer drugs in some cases includes exiting a DNA damage cell cycle arrest and entry into mitosis. Furthermore, some cells are able to survive this process at a time when the genome is most susceptible to change or rearrangement. Checkpoint adaptation is poorly characterised in human cells; we predict that increasing our understanding of this pathway may help to understand genomic instability in cancer cells and provide insight into methods to improve the efficacy of current cancer therapies.