There is an increasing use of computers in the design, manufacture and manipulation of physical objects. An important aspect of reasoning about such actions concerns the motion of objects in contact. The study of problems of this nature requires not only the ability to represent physical objects but the development of a framework or theory in which to reason about them. In this paper such a development is investigated and a fundamental theorem concerning the motion of objects in contact is proved. The simplest form of this theorem states that if two objects in contact can be moved to another configuration in which they are in contact, then there is a way to move them from the first configuration to the second configuration such that the objects remain in contact throughout the motion. This result is proved when translation and rotation of objects are allowed. The problem dealing with more generalized types of motion is also discussed. This study has obvious applications in compliant motion and in motion planning.