The achievement of controlled fusion will make available excess neutrons which can be used in various ways in combined fission-fusion power cycles. The various schemes are here classified as hybrid–in which heavy-element fission occurs in the fusion blanket; symbiotic–in which fissile material is bred in the absence of fission events; and augean–in which the neutron surplus is used to transmute fission reactor waste products. The earliest work in this field, unpublished and originally classified secret, is reviewed. Recent studies of hybrid systems based on subcritical thermal and sub-critical fast-fission blankets are described in detail. An analysis of symbiosis based on the Th-U fuel cycle is summarized. The conditions required for high-level waste transmutation are presented; Augean systems seem best suited to actinide waste burning. It is concluded that basic physics of various fission-fusion systems has been surveyed but the engineering and economic aspects of combined systems have been largely ignored. It remains to be shown whether fission-fusion systems will prove superior to alternative methods of accomplishing the same ends.