Perivascular spaces include a variety of passageways around arterioles, capillaries and venules in the brain, along which a range of substances can move. Although perivascular spaces were first identified over 150 years ago, they have come to prominence recently owing to advances in knowledge of their roles in clearance of interstitial fluid and waste from the brain, particularly during sleep, and in the pathogenesis of small vessel disease, Alzheimer disease and other neurodegenerative and inflammatory disorders. Experimental advances have facilitated in vivo studies of perivascular space function in intact rodent models during wakefulness and sleep, and MRI in humans has enabled perivascular space morphology to be related to cognitive function, vascular risk factors, vascular and neurodegenerative brain lesions, sleep patterns and cerebral haemodynamics. Many questions about perivascular spaces remain, but what is now clear is that normal perivascular space function is important for maintaining brain health. Here, we review perivascular space anatomy, physiology and pathology, particularly as seen with MRI in humans, and consider translation from models to humans to highlight knowns, unknowns, controversies and clinical relevance.