Attosecond extreme-ultraviolet pulses1 have a complex space–time structure2. However, at present, there is no method to observe this intricate detail; all measurements of the duration of attosecond pulses are, to some extent, spatially averaged1, 3, 4, 5. A technique for determining the full space–time structure would enable a detailed study of the highly nonlinear processes that generate these pulses as a function of intensity without averaging6, 7. Here, we introduce and demonstrate an all-optical method to measure the space–time characteristics of an isolated attosecond pulse. Our measurements show that intensity-dependent phase and quantum-path interference both play a key role in determining the pulse structure. In the generating medium, the attosecond pulse is strongly modulated in space and time. Propagation modifies but does not erase this modulation. Quantum-path interference of the single-atom response, previously obscured by spatial and temporal averaging, may enable measuring the laser-field-driven ion dynamics with sub-cycle resolution.