This paper builds up on the Capture Point concept and exploits the simple form of the dynamical equations of the Linear Inverted Pendulum model when formulated in terms of the center of mass and the Capture Point. The presented methods include (i) the derivation of a Capture Point (CP) control principle based on the natural dynamics of the linear inverted pendulum (LIP), which stabilizes the walking robot and motivates (ii) the design of a CP tracking and a CP end-of-step controller. The exponential stability of the CP control law is proven. Tilting is avoided by proper projection of the commanded zero moment point. The robustness of the derived control algorithms is analyzed analytically and verified in simulation and experiments.