The endosomal sorting complexes required for transport (ESCRTs) catalyse one of the most unusual membrane remodelling events in cell biology. ESCRT-I and ESCRT-II direct membrane budding away from the cytosol by stabilizing bud necks without coating the buds and without being consumed in the buds. ESCRT-III cleaves the bud necks from their cytosolic faces. ESCRT-III-mediated membrane neck cleavage is crucial for many processes, including the biogenesis of multivesicular bodies, viral budding, cytokinesis and, probably, autophagy. Recent studies of ultrastructures induced by ESCRT-III overexpression in cells and the in vitro reconstitution of the budding and scission reactions have led to breakthroughs in understanding these remarkable membrane reactions.