In this chapter we have introduced the paradigm `Limit Cycle Walking'. This paradigm has been used for some time by a group of bipedal robotics researchers, but the concept had not been properly defined before: Limit Cycle Walking is a nominally periodic sequence of steps that is stable as a whole but not locally stable at every instant in time. Limit Cycle Walking releases the unnecessary stability constraint of sustained local stability that is often applied in bipedal robots. It only requires the much less restrictive cyclic