People need energy, and current demands for cleaner energy to offset and ultimately avoid greenhouse-gas emissions. Nuclear fission are often described as neutron poor but power rich. Fission has the demonstrated ability to produce greenhouse-gas-free energy, but this process produces a large stream of radioactive nuclear waste that includes potentially bomb-grade plutonium.  Meanwhile, fusion is often described as neutron rich but power poor: although fusion has the theoretical ability to generate large amounts of clean energy, it is not currently an economically viable power source.  However, by using fusion reactions' excess neutrons to cause a fission reaction in a surrounding fissionable blanket, the combined hybrid fusion-fission process (hybrid nuclear power generation) has the potential to greatly multiply the energy produced and reduce the waste volume and radioactivity. [1,2]
The concept of hybrid fusion-fission was first introduced in the 1950's and reintroduced by Nobel laureate Hans bethe in the late 1970's. [1,2] The hybrid nuclear fuel cycle has three main components: deuterium, tritium, and a fissionable blanket. When deuterium and tritium undergo fusion, an alpha particle and a fast neutron are produced.  The 14-MeV neutron then escapes from the plasma and is captured by the fissionable blanket, generating several fissile nuclei from fertile ones, and enabling the hybrid reactor to be a net fuel source.  The blanket itself can be fissile (e.g. uranium, thorium) or fertile (a fertile material can be converted to a fissionable material by neutron bombardment).  Fertile blankets are currently the means of active disposal of demilitarized supplies of fissionables or commercial nuclear fuel and waste streams, including spent nuclear fuel, enriched fissile materials, and depleted uranium. This process does not require enrichment or reprocessing, which are both associated with nuclear weapons production.  For a more detailed description of the fusion, fission, and fusion-fission processes, please refer to Kates-Harbeck. 
Hybrid reactors are considered inherently safe because they allow fission blankets to be operated subcritically under all conditions and would likely be less susceptible to instabilities.  The fission reaction is driven by neutrons provided through fusion ignition events, so if the fusion process is shut off or disrupted, the fission stops nearly instantaneously - a chain reaction is physically impossible because the fusion and fission reactions are decoupled.  Although the fusion equipment required raises the initial cost of the reactor, the fusion hybrid has low fuel consumption and waste volume, which lowers the long-term cost of fuel cycles. It also provides a method to stretch the available uranium and thorium supplies because the fusion reactor can use their abundant isotopes, unlike pure fusion reactors, which can only use specific isotopes that are far less common. 
Unfortunately, despite the glowing theoretical evaluation, fusion-fission hybrid reactors do not currently exist. Practically, the technology is difficult to implement and early attempts required "recreating the conditions that drive the reaction in the Sun".  Thus, although they may have long term benefits, the sheer complexity and cost of creating the hybrid reactor has proven a formidable hurdle.  Constructing a fission reactor within a fusion reactor presents an immense engineering challenge. As technology in plasma containment has advanced over recent, people became more optimistic about hybrid power, but many of the problems that still plague hybrid fusion-fission lie in engineering the necessarily large, expensive, powerful machinery.  Typical strategies involve trapping the plasma with an intense magnetic field inside a tokamak, although as of 2009, the longest-lived fusion reaction lasted only a few seconds.  Yet a wealth of ongoing research exists to tackle the challenges inherent in the implementation of hybrid fusion-fission because pure fusion power is even more difficult, nuclear waste disposal is a pressing problem, and greenhouse-gas emissions ought to be reduced.  Several voices even challenge the idea that hybrid reactors would, indeed, be safer nuclear power options. Despite the physical impossibility of criticality accidents, greater dangers lie in other accidents, such as the failure to deliver sufficient cooling water to the core after shutting down a reactor.  Hybrid power is not widely considered sufficiently compelling to undergo such a high-cost project with likely applications restricted to removal of nuclear waste while larger-scale efforts focus on pure fusion reactors.
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 E. Gerstner, "The Hybrid Returns," Nature 460, 25 (2009).
 H. A. Bethe, "The Fusion Hybrid," Physics Today 32, No. 4, 44 (May 1979).
 J. Kates-Harbeck, "The Fusion-Fission Hybrid," Physics 241, Stanford University, Winter 2011.
 J. P. Freidberg and A. C. Kadak, "Fusion-Fission Hybrids Revisited," Nature Phys. 5, 370 (2009).