The geometry of parallelizable manifolds – that is, teleparallelism – is summarized in the language of local frame fields. Some problems in continuum mechanics that relate to the couple stresses that are produced in the bending and twisting of prismatic beams and wires are then discussed. It is then shown that by going to a higher dimensional analogue of the geometry that one uses for one-dimensional deformable objects, one is basically using the methods of teleparallelism in the context of the Cosserat approach to deformable bodies.