Motion vision provides essential cues for navigation and course control as well as for mate, prey, or predator detection. Consequently, neurons responding to visual motion in a direction-selective way are found in almost all species that see. However, directional information is not explicitly encoded at the level of a single photoreceptor. Rather, it has to be computed from the spatio-temporal excitation level of at least two photoreceptors. How this computation is done and how this computation is implemented in terms of neural circuitry and membrane biophysics have remained the focus of intense research over many decades. Here, we review recent progress made in this area with an emphasis on insects and the vertebrate retina.