In this paper, we study the passive dynamics of quadrupedal bounding, based on a simplified model of our Scout II quadruped robot. Surprisingly, numerical return map studies reveal that passive generation of a large variety of cyclic bounding motion is possible. Most strikingly, local stability analysis shows that the dynamics of the open loop passive system alone can confer stability of the motion. Stability improves at higher speeds, which is in agreement with recent results from biomechanics. These results can be used in developing a general control methodology for legged robots, resulting from the synthesis of feed-forward and feedback models that take advantage of the mechanical system, and might explain the success of simple, open loop bounding controllers on our experimental robot.