Abstract: This talk will present work by the speaker and colleagues on the systematic design of feedback controllers for walking and running motions in bipedal robots, with emphasis on the careful analysis of the stability properties of the closed-loop system. The presentation will provide a brief overview of past accomplishments on robots with rigid links and joints, and then move on to quite recent work on the control of robots with compliant elements. The talk is liberally illustrated with graphics and videos that explain and support the underlying mathematical theory.