A classification of a family of 3-revolute (3R) positioning manipulators is established. This classification is based on the topology of their workspace. The workspace is characterized in a half-cross section by the singular curves of the manipulator. The workspace topology is defined by the number of cusps and nodes that appear on these singular curves. The design parameters space is shown to be partitioned into nine subspaces of distinct workspace topologies. Each separating surface is given as an explicit expression in the DH-parameters.