Insect navigation has been a focus of behavioural study for many years, and provides a striking example of cognitive complexity in a miniature brain. We have used computational modelling to bridge the gap from behaviour to neural mechansims by relating the computational requirements of navigational tasks to the type of computation offered by invertebrate brain circuits. We have shown that visual memory of multiple views could be acquired by associative learning in the mushroom body neuropil, and allow insects to recapitulate long routes. We have also proposed a circuit in the central complex neuropil that integrates sky compass and optic flow information on an outbound path and can thus steer the animal directly home. The models are strongly constrained by neuroanatomy, and are tested in realistic agent and robot simulations.