Motivated by several physical systems, we study a simple model of driven, two-species lattice gases. Our system consists of only two types of NON-interacting (apart from an excluded volume constraint) particles, diffusing on a periodic lattice and with biased moves in opposite directions. On a square system, increasing the overall particle density leads to a transition - from a homogeneous phase with high particle current to one with spatial structure and minimal current. For rectangular cases, several structures can appear, with relative frequency depending on the aspect ratio of the system. Using a simple continuum theory, we are able to describe much of the novel transitions. Variations and generalizations, as well as the physical systems they model, will be discussed.