Currently, there is an increasing interest in heavy ion radiotherapy (RT) and a number of new facilities are being installed in Europe and Japan. This development is accompanied by intensive technical, physical, and clinical research. The authors identify six research fields where progress is likely and propose a thesis on the expected achievements for each of the fields: (1) Synchrotrons with active energy variation and three-dimensional beam scanning will be the standard in ion beam RT. (2) Common standards for precise measurement, prescription, and reporting of dose will be available. (3) Intensity-modulated particle therapy will be state-of-the-art. (4) Time-adaptive treatments of moving targets will be feasible. (5) Therapeutic effectiveness of heavy ions will be known for the most important indications while cost effectiveness will remain to be shown. (6) The potential of high-linear energy transfer radiation will be known. The rationale for each of these theses is described.