With their working mechanisms based on ion migration, the switching dynamics and electrical behaviour of memristive devices resemble those of synapses and neurons, making these devices promising candidates for brain-inspired computing. Built into large-scale crossbar arrays to form neural networks, they perform efficient in-memory computing with massive parallelism by directly using physical laws. The dynamical interactions between artificial synapses and neurons equip the networks with both supervised and unsupervised learning capabilities. Moreover, their ability to interface with analogue signals from sensors without analogue/digital conversions reduces the processing time and energy overhead. Although numerous simulations have indicated the potential of these networks for brain-inspired computing, experimental implementation of large-scale memristive arrays is still in its infancy. This Review looks at the progress, challenges and possible solutions for efficient brain-inspired computation with memristive implementations, both as accelerators for deep learning and as building blocks for spiking neural networks.