Motion planning for constrained systems is a version of the motion planning problem in which the motion of a robot is limited by constraints. For example, one can require that a humanoid robot such as a PR2 remain upright by constraining its torso to be above its base or require that an object such as a bucket of water remain upright by constraining the vertices of the object to be parallel to the robots base. Grasping can be modeled by requiring that the end effectors of the robot be located at specified handle positions. Constraints might require that the robot remain in contact with a surface, or that certain joints of the robot remain in contact with each other (e.g., closed chains). Such problems are particularly difficult because the constraints form a manifold in C-space, and planning must be restricted to this manifold. High-degree-of-freedom motion planning and motion planning for constrained systems has applications in parallel robotics, grasping and manipulation, computational biology and molecular simulations, and animation. We introduce a new concept, reachable volumes, that are a geometric representation of the regions the joints and end effectors of a robot can reach, and use it to define a new planning space called RV-space where all points automatically satisfy a problems constraints. Visualizations of reachable volumes can enable operators to see the regions of workspace that different parts of the robot can reach. Samples and paths generated in RV-space naturally conform to constraints, making planning for constrained systems no more difficult than planning for unconstrained systems. Consequently, constrained motion planning problems that were previously difficult or unsolvable become manageable and in many cases trivial. We introduce tools and techniques to extend the state-of-the-art sampling-based motion planning algorithms to RV-space. We define a reachable volumes sampler, a reachable volumes local planner, and a reachable volumes distance metric. We showcase the effectiveness of RV-space by applying these tools to motion planning problems for robots with constraints on the end effectors and/or internal joints of the robot. We show that RV-based planners are more efficient than existing methods, particularly for higher-dimensional problems, solving problems with 1000 or more degrees of freedom for multi-loop and tree-like linkages.