In a study of the kinematics of an Octopus's arm from the robotics point of view, we consider two models: a discrete model and a continuum ,nodel. Both models represent control of the curve in space defined by the centerline of the arm and the posi tion of the suction elements of the arm relative to the centerline. The two models also represent the control of the arm by means of three groups of muscles: the longitudinal, transverse and heli coidal groups. Another feature of the kinematics exhibited by the two models is the use of incompressibility in the control of the arm. In the discrete model, the arm is represented as a sequence of parallel mechanisms. Each mechanism in the sequence con sists of two parallel platforms of variable dimensions so that the resulting sequence of platforms represents the various cross sec tions of the arm. The continuum rod model that we use allows lin ear in-plane deformation of the cross-section and assumes that cross-sections remain perpendicular to the axis of the rod.