Physics-based simulation has emerged as a popular approach for realistic animation and analysis of rigid and articulated bodies in motion. This chapter briefly reviewed basic principles of unconstrained rigid body mechanics and then focused on the more challenging constrained rigid body mechanics principles. We have outlined basic concepts of rigid and articulated body modeling and simulation and advocated a constraint-based motion control that is based on motors implemented by constraints imposed on the position, velocity and/or acceleration of joint angles or points rigidly attached to the bodies. The formulated approach to control is simple and accurate within the context of readily available physics-based engines. That said, general control of complex articulated models, such as humanoids, is very challenging, especially in absence of trajectories that constrain all or most parts of the body over time. In particular, design of controllers that reproduce dynamics and energetics of human motion as well as can model dynamic variations due to the physical morphology or style of the individual remains an open issue. A variety of other approaches to motion control exist (see Section 1.1). For instance, task-based control (where the user specifies the task instead of joint angles or trajectories, e.g., pick up a mug from the table) has been emerging as the new alternative direction in the control and has a number of appealing properties from the point of view of animators and game designers. Discussing these alternative approaches falls outside the scope of this chapter. Lastly, we also do not consider numerical and performance aspects of the constraint-based motion control method and do not discuss various integration methods that clearly affect the quality (and the speed) of resulting simulations (see (Boeing et al., 2007) for discussion). That said, constraint-based motion control has become the standard approach for animating virtual worlds with stunning realism. This approach is versatile enough to model distinct phenomena like body articulation, joint actuation and contact in a uniform way; it is also capable of producing stable high quality simulations with predictable results in real time. Consequently, constraint-based control has become the default motion control strategy employed by all major commercial and open-source simulation packages.