We introduce a new concept, reachable volumes, that denotes the set of points that the end effector of a chain or linkage can reach. We show that the reachable volume of a chain is equivalent to the Minkowski sum of the reachable volumes of its links, and give an efficient method for computing reachable volumes. We present a method for generating configurations using reachable volumes that is applicable to various types of robots including open and closed chain robots, tree-like robots, and complex robots including both loops and branches. We also describe how to apply constraints (both on end effectors and internal joints) using reachable volumes. Unlike previous methods, reachable volumes work for spherical and prismatic joints as well as planar joints. Visualizations of reachable volumes can allow an operator to see what positions the robot can reach and can guide robot design. We present visualizations of reachable volumes for representative robots including closed chains and graspers as well as for examples with joint and end effector constraints.