Conversion of the long-lived nuclides of the elements plutonium, neptunium, americium and curium, generated during operation of nuclear reactors by neutron capture in U-238. Particularly for direct ultimate waste disposal of spent fuel elements, the partly very long half-lives of the nuclides Np-237, Pu-238, Pu-239, Pu-240, Am-241, Am-243, Cm-243 and Cm-244 emitting alpha radiation require proof of safe storage over very long periods. By nuclear conversions, either due to direct fission as for Pu-239 or Conversion in easily fissionable nuclides by neutron capture relatively short-lived or stable fission products are finally produced. For this purpose, it is necessary to reprocess the spent nuclear fuel, to separate these transuranium elements from the fission products and to convert them in suitable nuclear reactors using neutrons. In addition to the reactors as neutron source for the Conversion, subcritical arrangements driven by accelerators are discussed as a possibility of “burning” Pu and higher actinides. Since a high number of neutrons can be provided by a strong proton beam from an accelerator and the resulting spallations in a suitable target material, a good prerequisite exists for such transmutation machines. As no self-perpetuating chain reaction takes place in such facilities, advantages with regard to safety behaviour are expected.