The workspace topologies of a family of 3-revolute (3R) positioning manipulators are enumerated. The workspace is characterized in a half-cross section by the singular curves. The workspace topology is defined by the number of cusps that appear on these singular curves. The design parameters space is shown to be divided into five domains where all manipulators have the same number of cusps. Each separating surface is given as an explicit expression in the DH-parameters. As an application of this work, we provide a necessary and sufficient condition for a 3R orthogonal manipulator to be cuspidal, i.e. to change posture without meeting a singularity. This condition is set as an explicit expression in the DH parameters.