Most solid tumours are aneuploid and many frequently mis-segregate chromosomes. This chromosomal instability is commonly caused by persistent mal-oriented attachment of chromosomes to spindle microtubules. Chromosome segregation requires stable microtubule attachment at kinetochores, yet those attachments must be sufficiently dynamic to permit correction of mal-orientations. How this balance is achieved is unknown, and the permissible boundaries of attachment stability versus dynamics essential for genome stability remain poorly understood. Here we show that two microtubule-depolymerizing kinesins, Kif2b and MCAK, stimulate kinetochore-microtubule dynamics during distinct phases of mitosis to correct mal-orientations. Few-fold reductions in kinetochore-microtubule turnover, particularly in early mitosis, induce severe chromosome segregation defects. In addition, we show that stimulation of microtubule dynamics at kinetochores restores stability to chromosomally unstable tumour cell lines, establishing a causal relationship between deregulation of kinetochore-microtubule dynamics and chromosomal instability. Thus, temporal control of microtubule attachment to chromosomes during mitosis is central to genome stability in human cells.