When using accelerator beams for cancer therapy, the three-dimensional freedom afforded by a gantry helps the treatment planner to spread out surface doses, avoid directions that intercept vital organs and irradiate a volume that is conformal with the tumour. The general preference is for an iso-centric gantry turning 360q in the vertical plane around the patient bed with sufficient space to be able to orientate the patient through 360q in the horizontal plane. For hadrontherapy, gantries are impressive structures of the order of 10 m in diameter and 100 tons in weight and to date only proton gantries have been demonstrated to operate satisfactorily. The increased magnetic rigidity of say carbon ions will make ion gantries more difficult and costly to build. For this reason, exo-centric gantries and, in particular the so-called 'Riesenrad' gantry with a single 90q bending magnet, merit further attention. The power consumption is reduced and the heavy magnets with their counterbalance weight are reduced and are kept close to the axis. The treatment room, which is lighter, is positioned at a larger radius, but only the patient bed requires careful alignment. An optics module called a 'rotator' is needed to match an incoming dispersion vector to the gantry in order to have an achromatic beam at the patient. A practical design is described that assumes the beam is derived from a slow-extraction scheme in a synchrotron and that the beam sizes are controlled by modules in the transfer line. Magnetic scanning is integrated into the gantry optics for both transverse directions.