A theory of quadrupedal locomotory control is proposed and tested on six planar animal models spanning nearly three orders of magnitude in body size:​ two horse models (676 Kg, 134 Kg), a goat model (25.2 Kg), two dog models (23.9 Kg, 5.09 Kg) and a chipmunk model (0.115 Kg). To capture the significant mechanical characteristics of quadrupedal running, each model is morphologically realistic with back and neck joints for modeling trunk and head flexibilities, and four flexible legs for modeling footfall patterns in galloping. In numerical simulation, the models both trot and gallop without having to actively balance. Control methods are developed which stabilize body pitch without requiring sensory feedback of body pitch. The controller applies hip and shoulder torques to move each foot at a constant tangential velocity relative to the trunk, like a rim of a steadily rolling wheel. If the tangential forelimb target velocity is smaller than a model's forward running speed, and the hindlimb target velocity is greater, the shoulder generally applies a braking torque during stance, and the hip a thrusting torque. This behavior increases model stability by decreasing angular fluctuations in pitch, keeping each model trunk more parallel to the ground. Passive springs are used in the back, neck, and legs, enabling a model to rebound efficiently from the ground during each step. In Chapter 2, the horse model (134 Kg) successfully predicts animal stiffness, limb excursion angle, relative stride length, stride frequency, and the metabolic cost of transport at seven different running speeds ranging from a slow trot to a fast gallop. In Chapter 3, the six animal models successfully predict how body size affects the mechanics and energetics of quadrupedal running.