Today, two basically different methods have been developed for production of accelerated radioactive ion beams (RIB). The fragmentation of intense heavy–ion beams, in which the forward momentum imparted to the primary beam fragments is preserved and exploited for mass separation, study and further reactions, is the preferred technique at present. The second method, which is the subject of this review, makes use of spallation, fission and fragmentation reactions in thick targets driven by light particles from a first accelerator or a nuclear reactor. The radioactivity produced is brought to rest in the target and then has to be separated and transformed into an ion beam in order to be post–accelerated in a second machine. This method has in the past 30 years been used successfully at many on–line mass separators to produce low–energy radioactive ion beams. Techniques for the transfer of the nuclear reaction products into an ion beam have been optimized with respect to the individual physical and chemical properties of 70% of the chemical elements. Several new ideas in efficiently matching such an on–line mass separator as injector to a heavy–ion accelerator are currently being developed so that this method holds much promise for the future, in particular when it comes to selecting the intensity and energy choice of the secondary beams. This paper is devoted to a systematic description and discussion of this two–accelerator type of RIB facility, of which many are currently in various phases of construction and planning.