Animals can swerve, dodge, dive, climb, turn and stop abruptly. Their stability and maneuverability are remarkable, but a challenge to quantify. Formal stability analysis can allow for quantitative comparisons within and among species. Stability analysis used in concert with a template (a simple, general model that serves as a guide for control) can lead to testable hypotheses of function. Neural control models postulated without knowledge of the animal's mechanical (musculo-skeletal) system can be counterproductive and even destabilizing. Perturbations actively corrected by reflex feedback in one direction can result in perturbations in other directions because the system is coupled dynamically. The passive rate of recovery from a perturbation in one direction differs from rates in other directions. We hypothesize that animals might exert less neural control in directions that rapidly recover via passive dynamics (e.g., in body orientation and rotation). By contrast, animals are likely to exert more neural control in directions that recover slowly or not at all via passive dynamics (e.g., forward velocity and heading). Neural control best enhances stability when it works with the natural, passive dynamics of the mechanical system. Measuring maneuverability is more challenging and new, general metrics are needed. Templates reveal that simple analyses of summed forces and quantification of the center of pressure can lead to valuable hypotheses, whereas kinematic descriptions may be inadequate. The study of stability and maneuverability has direct relevance to the behavior and ecology of animals, but is also critical if animal design is to be understood. Animals appear to be grossly over-built for steady-state, straight-ahead locomotion, as they appear to possess too many neurons, muscles, joints and even too many appendages. The next step in animal locomotion is to subject animals to perturbations and reveal the function of all their parts.