In the last decade the outlines of the neural structures subserving the sense of direction have begun to emerge. Several investigations have shed light on the effects of vestibular input and visual input on the head direction representation. In this paper, a model is formulated of the neural mechanisms underlying the head direction system. The model is built out of simple ingredients, depending on nothing more complicated than connectional specificity, attractor dynamics, Hebbian learning, and sigmoidal nonlinearities, but it behaves in a sophisticated way and is consistent with most of the observed properties of real head direction cells. In addition it makes a number of predictions that ought to be testable by reasonably straightforward experiments.