In the attempt to build adaptive and intelligent machines, roboticists have looked at neuroscience for more than half a century as a source of inspiration for perception and control. More recently, neuroscientists have resorted to robots for testing hypotheses and validating models of biological nervous systems. Here, we give an overview of the work at the intersection of robotics and neuroscience and highlight the most promising approaches and areas where interactions between the two fields have generated significant new insights. We articulate the work in three sections, invertebrate, vertebrate and primate neuroscience. We argue that robots generate valuable insight into the function of nervous systems, which is intimately linked to behaviour and embodiment, and that brain-inspired algorithms and devices give robots life-like capabilities.