A method is developed for the three-dimensional nonlinear motions of marine risers, pipelines and offshore loading towers. The technique is based on the finite element method and the separation of the rigid body motions and deformations of elements under conditions of finite rotations. The position of the riser is defined by a set of co-rotational axes and the evaluation of the finite rotations of this convected system is detailed. The method includes all nonlinear effects including geometry changes, bending-axial and bending-torsional coupling and follower forces and pressures. The computation of Morison-type loads is described with respect to three-dimensional axes. In order to verify the procedure numerical results are presented for the case of a vertical cantilever under bi-directional loading and axial torque; the results are shown to agree exactly with independent calculations. Results are also presented for an example from the literature on the three-dimensional response of an articulated tower to noncollinear waves and current.