The challenge of fabricating bio-inspired hardware is building ultra-high density networks out of complex processing units interlinked by tunable connections. Nanodevices exploiting spin electronics (or spintronics) can be a key technology in this context. In particular, magnetic tunnel junctions are well suited for this purpose because of their multiple and tunable functionalities. One such functionality, non-volatile memory, can provide massive embedded memory in unconventional circuits, thus escaping the von-Neumann bottleneck arising when memory and processors are located separately. Other functionalities of spintronic devices that could be beneficial for bio-inspired computing include tunable fast non-linear dynamics, controlled stochasticity, and the ability of single devices to adopt different functionalities in different operating conditions. Large networks of interacting spintronic nano-devices can have their interactions tuned to induce complex dynamics such as synchronization, chaos, soliton diffusion, phase transitions, criticality, and convergence to multiple metastable states. A number of groups have recently proposed bio-inspired architectures that include one or several types of spin torque nanodevices. In this article we show how spintronics can be used for bio- inspired computing. We review the different approaches that have been proposed, the recent advances in this direction, and the challenges towards fully integrated spintronics-CMOS (Complementary metal oxide semiconductor) bio-inspired hardware.