An approach to the analysis of swept volumes is introduced. It is shown that every smooth Euclidean motion or sweep, can be identified with a first-order, linear, ordinary differential equation. This sweep differential equation provides useful insights into the topological and geometrical nature of the swept volume of an object. A certain class, autonomous sweeps, is identified by the form of the associated differential equation, and several properties of the swept volumes of the members of this class are analyzed. The results are applied to generate swept volumes for a number of objects. Implementation of the sweep differential equation approach with computer-based numerical and graphical methods is also discussed.