Intermittent hypoxia, reoxygenation, and hypercapnia or hypocapnia occur in both adults and children during untreated apnoea and hypopnoea, along with changes in cerebral blood flow and sleep fragmentation. These effects can result in cognitive deficits with functional effects on work and school efficiency. The assessment of how obstructive sleep apnoea affects cognition depends on the specificity and sensitivity of the tests, which are rarely developed specifically for obstructive sleep apnoea. In this Review, we discuss both the neural adaptive and maladaptive processes in response to hypoxaemia. The net result on cognitive and emotional performance depends on the stage of this dynamic process, effects on other body systems, cognitive reserve, and idiosyncratic susceptibility. We also explore the contribution of fragmented sleep, and the disruption of sleep structure, with focus on the effect at different times in the development of disease. This Review will address the gap in the underlying pathophysiology of new clinical and translational findings, and argue their contribution to the inherent complexity of the association between obstructive sleep apnoea and the brain.