We present a flexible and efficient approach for generating multilegged locomotion. Our model-predictive control (MPC) system efficiently generates terrain-adaptive motions, as computed using a three-level planning approach. This leverages two commonly-used simplified dynamics models, an inverted pendulum on a cart model (IPC) and a centroidal dynamics model (CDM). Taken together, these ensure efficient computation and physical fidelity of the resulting motion. The final full-body motion is generated using a novel momentum-mapped inverse kinematics solver and is responsive to external pushes by using CDM forward dynamics. For additional efficiency and robustness, we then learn a predictive model that then replaces two of the intermediate steps. We demonstrate the rich capabilities of the method by applying it to monopeds, bipeds, and quadrupeds, and showing that it can generate a very broad range of motions at interactive rates, including banked variable-terrain walking and running, hurdles, jumps, leaps, stepping stones, monkey bars, implicit quadruped gait transitions, moon gravity, push- responses, and more.