Animal cells undergo dramatic changes in shape, mechanics and polarity as they progress through the different stages of cell division. These changes begin at mitotic entry, with cell-substrate adhesion remodelling, assembly of a cortical actomyosin network and osmotic swelling, which together enable cells to adopt a near spherical form even when growing in a crowded tissue environment. These shape changes, which probably aid spindle assembly and positioning, are then reversed at mitotic exit to restore the interphase cell morphology. Here, we discuss the dynamics, regulation and function of these processes, and how cell shape changes and sister chromatid segregation are coupled to ensure that the daughter cells generated through division receive their fair inheritance.