Nuclear transmutation is the conversion of one chemical element or isotope into another, which occurs through nuclear reactions. Natural transmutation occurs when radioactive elements spontaneously decay over a long period of time and transform into other more stable elements. Artificial transmutation occurs in machinery that has enough energy to cause changes in the nuclear structure of the elements. Machines that can cause artificial transmutation include particle accelerators and tokamak reactors as well as conventional fission power reactors. Nuclear transmutation is considered as a possible mechanism for reducing the volume and hazard of radioactive waste.
The term transmutation dates back to the search for the philosopher's stone. In alchemy, it was believed that the transformation of base metals into gold could be accomplished in table-top experiments. The alchemical belief in transmutation was based on a thoroughly wrong understanding of the underlying processes. The belief in transmutation continued to be widespread in Europe until Antoine Lavoisier, in the 18th century, replaced the alchemical theory of elements with the modern theory of chemical elements and when John Dalton further developed the notion of atoms to explain various chemical processes. The disintegration of atoms is a distinct process involving much greater energies than could be achieved by alchemists.
It was first consciously applied to modern physics by Frederick Soddy when he, along with Ernest Rutherford, discovered that radioactive thorium was converting itself into radium in 1901. At the moment of realization, Soddy later recalled, he shouted out: "Rutherford, this is transmutation!" Rutherford snapped back, "For Christ's sake, Soddy, don't call it transmutation. They'll have our heads off as alchemists."
Later in the twentieth century the transmutation of elements within stars was elaborated, accounting for the relative abundance of elements in the universe. In their 1957 paper Synthesis of the Elements in Stars, William Alfred Fowler, Margaret Burbidge, Geoffrey Burbidge, and Fred Hoyle explained how the abundances of essentially all but the lightest chemical elements could be explained by the process of nucleosynthesis in stars.
Author Ken Croswell summarised their discoveries thus:
Burbidge, Burbidge, Fowler, Hoyle
Took the stars and made them toil:
Carbon, copper, gold, and lead
Formed in stars, is what they said
Ironically, it transpired that, under true nuclear transmutation, it is far easier to turn gold into lead than the reverse reaction, which was the one the alchemists had ardently pursued. Nuclear experiments have successfully transmuted lead into gold, but the expense far exceeds any gain. It would be easier to convert gold into lead via neutron capture and beta decay by leaving gold in a nuclear reactor for a long period of time.
More information on gold synthesis, see Synthesis of noble metals.
Transmutation of transuranium elements (actinides) such as the isotopes of plutonium, neptunium, americium, and curium has the potential to help solve the problems posed by the management of radioactive waste, by reducing the proportion of long-lived isotopes it contains. When irradiated with fast neutrons in a nuclear reactor, these isotopes can be made to undergo nuclear fission, destroying the original actinide isotope and producing a spectrum of radioactive and nonradioactive fission products.
For instance, plutonium can be reprocessed into MOX fuels and transmuted in standard reactors. The heavier elements could be transmuted in fast reactors, but probably more effectively in a subcritical reactor which is sometimes known as an energy amplifier and which was devised by Carlo Rubbia. Fusion neutron sources have also been proposed as well suited. 
There are several fuels that can incorporate plutonium in their initial composition at Beginning of Cycle (BOC) and have a smaller amount of this element at the End of Cycle (EOC). During the cycle, plutonium can be burnt in a power reactor, generating electricity. This process is not only interesting from a power generation standpoint, but also due to its capability of consuming the surplus weapons grade plutonium from the weapons program and plutonium resulting of reprocessing Spent Nuclear Fuel (SNF).
Mixed Oxide fuel (MOX) is one of these. Its blend of oxides of plutonium and uranium constitutes an alternative to the Low Enriched Uranium (LEU) fuel predominantly used in Light Water Reactors (LWR). Since uranium is present in MOX, although plutonium will be burnt, second generation plutonium will be produced through the radiative capture of U-238 and the two subsequent beta minus decays.
Fuels with plutonium and thorium are also an option. In these, the neutrons released in the fission of plutonium are captured by Th-232. After this radiative capture, Th-232 becomes Th-233, which undergoes two beta minus decays resulting in the production of the fissile isotope U-233. The radiative capture cross section for Th-232 is more than three times that of U-238, yielding a higher conversion to fissile fuel that that from U-238. Due to the absence of uranium in the fuel, there is no second generation plutonium produced, and the amount of plutonium burnt will be higher than in MOX fuels. However, U-233, which is fissile, will be present in the SNF. Weapons-grade and reactor-grade plutonium can be used in plutonium-thorium fuels, being weapons-grade plutonium the one that shows a bigger reduction in the amount of Pu-239.
Isotopes of plutonium and other actinides tend to be long-lived with half-lives of many thousands of years, whereas radioactive fission products tend to be shorter-lived (most with half-lives of 30 years or less). From a waste management viewpoint, transmutation of actinides eliminates a very long-term radioactive hazard and replaces it with a much shorter-term one.
It is important to understand that the threat posed by a radioisotope is influenced by many factors including the chemical and biological properties of the element. For instance caesium has a relatively short biological halflife (1 to 4 months) while strontium and radium both have very long biological half-lives. As a result strontium-90 and radium are much more able to cause harm than caesium-137 when a given activity is ingested.
Many of the actinides are very radiotoxic because they have long biological half-lives and are alpha emitters. In transmutation the intention is to convert the actinides into fission products. The fission products are very radioactive, but the majority of the activity will decay away within a short time. The most worrying shortlived fission products are those that accumulate in the body, such as iodine-131 which accumulates in the thyroid gland, but it is hoped that by good design of the nuclear fuel and transmutation plant that such fission products can be isolated from humans and their environment and allowed to decay. In the medium term the fission products of highest concern are strontium-90 and caesium-137; both have a half life of about 30 years. The caesium-137 is responsible for the majority of the external gamma dose experienced by workers in nuclear reprocessing plants and at this time (2005) to workers at the Chernobyl site. When these medium-lived isotopes have decayed the remaining isotopes will pose a much smaller threat.
Some radioactive fission products can be converted into shorter-lived radioisotopes by transmutation. Transmutation of all fission products with halflife greater than one year is studied in , with varying results.
Sr-90 and Cs-137, with halflives of about 30 years, are the largest radiation emitters in used nuclear fuel on a scale of decades to a few hundreds of years, and are not easily transmuted because they have low neutron absorption cross sections. Instead, they should simply be stored until they decay. Given that this length of storage is necessary, the fission products with shorter halflives can also be stored until they decay.
The next longer-lived fission product is Sm-151, which has a halflife of 90 years, and is such a good neutron absorber that most of it is transmuted while the nuclear fuel is still being used; however effectively transmuting the remaining Sm-151 in nuclear waste would require separation from other isotopes of samarium. Given the smaller quantities and its low-energy radioactivity, Sm-151 is less dangerous than Sr-90 and Cs-137 and can also be left to decay.
Finally, there are 7 long-lived fission products. They have much longer halflives in the range 211,000 years to 16 million years. Two of them, Tc-99 and I-129, are mobile enough in the environment to be potential dangers, are free or mostly free of mixture with stable isotopes of the same element, and have neutron cross sections that are small but adequate to support transmutation. Also, Tc-99 can substitute for U-238 in supplying Doppler broadening for negative feedback for reactor stability.  Most studies of proposed transmutation schemes have assumed 99Tc, 129I, and tansuranics as the targets for transmutation, with other fission products, activation products, and possibly reprocessed uranium remaining as waste. 
Of the remaining 5 long-lived fission products, Se-79, Sn-126 and Pd-107 are produced only in small quantities (at least in today's thermal neutron, U-235-burning light water reactors) and the last two should be relatively inert. The other two, Zr-93 and Cs-135, are produced in larger quantities, but also not highly mobile in the environment. They are also mixed with larger quantities of other isotopes of the same element.