The robust behaviours, specialised sensors and small brains of insects have been a source of inspiration for efficient processing algorithms for sensorimotor control in robots.  But insects are not just reactive systems; rather they are capable of much more complex behaviours - including learning, integration of multimodal cues, real-world navigation and flexible behavioural choice - than any existing autonomous robots. I will review some of these capabilities, what is known of the neural circuits that underly them, and how we have tested our understanding of these circuits by implementing them in simulated and robot systems.