Neuromorphic computing emulates a biological brain at different levels of the computer hierarchy by exploiting brain‐inspired principles in designing novel devices, algorithms, and architectures. It is believed to have a lower power budget and a higher efficiency in performing cognitive tasks, and is arguably the most promising approaches for next‐generation artificial intelligence. Despite the potentials, progress of neuromorphic computing is constrained by a lack of dedicated hardware. Spintronics is a fast‐evolving discipline that exploits the spin degree of freedom in electronics. The interplay of magnetic and electrical properties of spintronic devices gives rise to a wide range of amazing phenomena, which are both intrinsically conducive to neuromorphic computing and highly compatible with conventional manufacturing technologies. Here, the development of neuromorphic computing with reference to spintronics is reviewed. The state‐of‐the‐art spintronic technologies, such as the magnetic tunnel junction, spin–orbit torque, domain wall propagation, magnetic skyrmions, and antiferromagnet, are highlighted and how they can used for artificial neurons and synapses in different artificial neural networks are discussed. The technical challenges to be overcome for realizing more powerful all‐spin artificial neural networks are also evaluated.