AI systems managed to reach and even exceed human performance in various cognitive tasks, ranging from image recognition to strategic games and to reasoning. Yet, today’s computing systems based on the classical von-Neumann architecture dating from the 1940s cannot efficiently address these highly data-intensive workloads. It is becoming increasingly clear that we need to transition to non-von Neumann architectures in which memory and logic co-exist in some form. Brain-inspired computing is one such approach, where inspiration is taken from biological observations of the brain. One can build in-memory computing units where the separation between memory and processing is blurred, and physical attributes and state dynamics of memory devices are exploited to perform certain computational tasks. The neural structure and operation of the brain including the rich neural and synaptic dynamics can also be adopted to add to the information processing abilities and improve the efficiency of computing systems. Resistive memory is expected to play a key role for brain-inspired computing, from building in-memory computing arrays to emulating neurons and synapses for neuromorphic computing. Explorations in device technology and memory architectures could further enhance the capabilities of brain-inspired computing systems.