Since Hans Bethe's seminal paper in 1979, fission-fusion hybrid reactors have taken their place in the realm of science fiction, a novel idea that simply wouldn't be practical enough to implement. Bethe explained the advantages of the hybrid reactor beautifully: The fusion reaction (for example, between deuterium and tritium) will produce excess neutrons. Each excess neutron (traveling at an energy of approximately 14 MeV) will either yield two more neutrons off a uranium or thorium nucleus, or four neutrons in a uranium or thorium fission reaction. "Because the abundant isotopes U-238 and Th-232 are used, all of the uranium and thorium will be available for power production, not just a fraction of a percent as at present in light-water reactors. Much lower grade uranium and thorium ores can therefore be used than at present". 
However, because fusion has been slow to develop, the fission-fusion hybrid that Bethe proposed has not become a significant reality. However, as the challenges we face today grow in magnitude exponentially, such as the issue of nuclear waste, the fission-fusion hybrid is getting a second look.
In the last several years, many scientists have suggested using fission-fusion hybrids to deal with the issue of nuclear waste by transmutation - what's known as a Fusion-fission Transmutation System (FFTS).  This work builds off of the previously established knowledge of transmutation with fusion neutrons. When dealing with nuclear the problem often is dealing with isotopes that are very unstable but have very long half-lives, as is often the case with minor actinides that form when faster neutrons collide with Plutonium nuclei. In this case, the Pu nucleus does not undergo fission, and instead undergoes transmutation to a higher actinide.  The end result is (after all of the U-233 and Pu-239 is burned) about 750 kg of radio isotopes, 50 kg of which have very long half-lives. In these cases, fusion neutrons are used to transmute these radioactive fission products, such as Tc-99 and Cs-137. In the case of Tc-99 (an isotope with a half-life of 200,000 years), and slow neutron will be absorbed by the nucleus (which has a cross-section of 20 barns) thus forming Tc-100. This quickly decays into Ru-100, which is stable.
Knowing this, the proposed FFTS takes this one step further. The key in the FFTS is the use of a high density compact fusion neutron source (CFNS) produces fast neutrons which augment the rate of nuclear reactions. The CFNS will be encapsulated by a blanket of fissile material; as the reaction takes place and the fissile blanket becomes more depleted, more transuranic elements are now part of this subcritical blanket. These transuranic elements will now undergo transmutation, either transmuting into fissile material or stable isotopes. This is a significant improvement over Light Water Reactors, in which these long-lived biologically hazardous isotopes cannot undergo fission.  Moreover, traditional ways of dealing with hazardous waste, such as fast-spectrum reactors (critical FRs), still do not deal with this biohazardous "sludge."
Again, the key is the CFNS, which has to be compact, but also capable of producing high neutron flux. If a CFNS is designed to be small enough to fit inside a blanket of fissile material but powerful enough to address the need for transmutation, then the fission-fusion hybrid will significantly improve nuclear waste management.
© Firas Abuzaid. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
 H. Bethe, "The Fusion Hybrid," Physics Today 32, No. 5, 44 (1979).
 M. Kotschenreuther, P. M. Valanjua, S. M. Mahajan and E. A. Schneider, "Fusion-Fission Transmutation Scheme - Efficient Destuction of Nuclear Waste," Fusion Engineering and Design 84, 83 (2009).
 W. Manheimer, "The Fusion Hybrid as a Key to Sustainable Development," J. Fusion Energy 23, 223 (2004).
 E. A. Schneider et al., "Burnup Simulations and Spent Fuel Characteristics of ZrO2 Based Inert Matrix Fuels," Nucl. Mater. 361, 41 (2007).