The Lemke-Howson algorithm for bimatrix games provides both an elementary proof of the existence of equilibrium points and an efficient computational method for finding at least one equilibrium point. The first half of this paper presents a geometrical view of the algorithm that makes its operation especially easy to visualize. Several illustrations are given, including Wilson’s example of “inaccessible” equilibrium points. The second half presents an orientation theory for the equilibrium points of (nondegenerate) bimatrix games and the Lemke-Howson paths that interconnect them; in particular, it is shown that there is always one more “negative” than “positive” equilibrium point.