Neuromorphic computing requires electronic systems that can perform massively parallel computational tasks with low energy consumption. Such systems have traditionally been based on complementary metal-oxide-semiconductor circuits, but further advances in computational performance will probably require devices that can offer high-order complexity combined with area and energy efficiency. Halide perovskites can handle both ions and electronic charges, and could be used to create adaptive computing systems based on intrinsic device dynamics. The materials also offer exotic switching phenomena, providing opportunities for multimodal systems. Here we explore the development of neuromorphic hardware systems based on halide perovskites. We examine how devices based on these materials can serve as synapses and neurons, and can be used in neuromorphic computing networks. We also consider the challenges involved in developing practical perovskite neuromorphic systems, and highlight how these systems could augment and complement digital circuits. This Review examines the development of neuromorphic hardware systems based on halide perovskites, considering how devices based on these materials can serve as synapses and neurons, and can be used in neuromorphic computing networks.