Continuum manipulator compliance enables operation in delicate environments at the cost of challenging actuation and control. In the case of catheter ablation of atrial fibrillation, the compliance of the continuum backbone lends an inherent safety to the device. This inherent safety frustrates attempts at precise, accurate, and fast control, limiting these devices to simple, static positioning tasks. This dissertation develops Interleaved Continuum-Rigid Manipulation, by which the hysteretic nonlinearities encountered in tendon-actuated continuum manipulators are compensated by discrete rigid joints located between continuum sections. The rigid joints introduce actuation redundancy, which an interleaved controller may use to avoid continuum nonlinearities and dynamic excitations, or to prefer particular configurations that may improve task accuracy, permit greater end-effector forces, or avoid environment obstacles. Two experimental systems explore the potential of these joints to 1) correct for actuation nonlinearities and enhance manipulator performance and 2) increase the manipulator’s dexterous workspace. These experiments expose important design and control observations that were not apparent in the general robotic and continuum literature.