The spatial mapping theory of hippocampal function proposes that the rat hippocampus is specialized for navigational computations, computations that allow the animal to solve difficult spatial problems. In this paper, we review evidence obtained by recording place cells and other "spatially tuned" cells from freely moving rats. Our main conclusion is that the nature of the signals carried by these cells and the ways in which the signals transform after changing the environment imply that the hippocampus and associated structures are able to represent aspects of the geometry of the environment.