We introduce a new method for synthesizing kinematic relationships for a general class of continuous backbone, or continuum , robots. The resulting kinematics enable real-time task and shape control by relating workspace (Cartesian) coordinates to actuator inputs, such as tendon lengths or pneumatic pressures, via robot shape coordinates. This novel approach, which carefully considers physical manipulator constraints, avoids artifacts of simplifying assumptions associated with previous approaches, such as the need to fit the resulting solutions to the physical robot. It is applicable to a wide class of existing continuum robots and models extension, as well as bending, of individual sections. In addition, this approach produces correct results for orientation, in contrast to some previously published approaches. Results of real-time implementations on two types of spatial multisection continuum manipulators are reported.