Continuum manipulators are a class of robots with many degrees-of-freedom, leading to highly flexible motion with inherent compliance. These attributes make them well suited for manipulation tasks and physical interaction with the environment. A high impact yet challenging field for exploring continuum robot designs is free-floating underwater manipulation with a remotely operated vehicle (ROV). In this article, we propose a modular, reconfigurable, cable-driven continuum arm for free-floating underwater manipulation and present a corresponding kinematics, control, and computation framework. The mechanical design consists of a continuum arm, an actuation unit, and a waterproof enclosure. The kinematics model is introduced as two mappings between three spaces: the joint space, the configuration space, and the task space. The differential kinematics for each mapping is also derived. An electronics system design is proposed for underwater applications, including the communication framework between the topside computer (above surface), on-board computer, and manipulator mechatronics. Experimental validation is presented to demonstrate the robot’s underwater functionality, test the limits of its articulation, and evaluate the arm’s stiffness. Future work includes field testing with an ROV platform and development of advanced controls and planning for manipulation tasks.