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This dissertation addresses the problem of control and kinematic planning for constrained robot systems. An example of a system of this type is a multifingered robot hand grasping an object. The individual fingers act as robot manipulators and are constrained by their contact with the object. If the contacts allow rolling between the object and the fingertips, it is possible for the constraints to be nonholonomic. That is, the constraints may not restrict the reachable configurations of the system, but rather, constrain only the allowable velocities of the system. <p> Using the multifingered hand as a motivating example, this dissertation presents a detailed analysis of the kinematics, dynamics, and control of robot systems with contact constraints. In particular, it presents a unified derivation of the dynamics of robot manipulators with Pfaffian velocity constraints, including the nonholonomic case. This derivation allows control laws to be specified which are provably stable for an entire class of systems, including unconstrained robots, robot hands, and other systems of multiple robots performing a coordinated task. A method for building complex controllers which respects this class of constraints is also developed using a set of simple primitives which allow hierarchical control structures to be created in an organized fashion. <p> Finally, the nonholonomic motion planning problem is introduced and discussed in detail. Using tools from differential geometric control theory, it is possible to classify and analyze systems with nonholonomic constraints. A brief review of the necessary tools along with a review of the current literature is presented. A practical method for steering nonholonomic systems using sinusoids is derived and applied to several kinematic systems with contact constraints.