The kinematics of the octopus’s arm is studied from the point of view of robotics. A continuum three-dimensional kinematic model of the arm, based on a nonlinear rod theory, is proposed. The model enables the calculation of the strains in various muscle fibers that are required in order to produce a given configuration of the arm—a solution to the inverse kinematics problem. The analysis of the forward kinematics problem shows that the strains in the muscle fibers at two distinct points belonging to a cross section of the arm determine the curvature and the twist of the arm at that cross section. The octopus’s arm lacks a rigid skeleton and the role of material incompressibility in enabling the configuration control is studied.