The best-known incarnation of this is the tokamak, first built in the Soviet Union in the 1950s. In a tokamak, deuterium and tritium are injected into a doughnut, or torus-shaped chamber, and heated to the point at which its electrons break free. Magnetic fields running along the centre contain and squeeze the plasma, and the high temperatures within the plasma then facilitate fusion. However, even the best tokamak designs – including the Joint European Torus (JET) in the United Kingdom and the Tokamak Fusion Test Reactor (TFTR) in the United States – haven't broken the barrier of making more energy than is required to keep the plasma hot and trapped. Much hope is being placed on Iter (International Thermonuclear Experimental Reactor), a €15 billion ($22 billion) project designed to build the world’s largest tokamak in the south of France. Iter is expected to commence operation at the end of this decade, with the first proper fusion tests scheduled for 2028. But it has been dogged by logistical issues – last month the US Senate spending panel voted to stop contributions to the project.
Many think another method called inertial confinement provides the best hope for a workable fusion reactor. This uses bombardment by high-energy photons – X-rays – to confine and compress a pellet of hydrogen and its isotopes. Successive X-ray pulses emanate from a large number of lasers completely surrounding the pellet, doing the work of heating, ionising, and compressing the hydrogen to the point where it can fuse. The biggest barrier to a working model lies in the X-ray lasers, which require a lot of energy to operate, but researchers at laboratories such as SLAC in the United States and the European X-ray Free Electron Laser are actively working to solve the problem.
The Z Machine at Sandia National Laboratory in the United States is a hybrid between magnetic and inertial confinement. Though the Z Machine itself is not a fusion reactor (and in fact is partly used for developing models for nuclear weapons), the powerful magnetic fields and X-rays it produces are part of a project in which a pellet of hydrogen is repeatedly bombarded with intense pulses of light to compress it.
There are other methods for confinement and compression, and more will no doubt be developed. Another path is aimed at lowering the energy barrier to deuterium fusion: forming molecules using muons instead of electrons. Muons are the heavier, unstable cousins of electrons. (We all have family members like that, I think.) Their presence in a molecule of two deuterium atoms brings the nuclei much closer together, consequently making fusion much more likely. However, once the energy cost of making muons and creating molecules using them is added in, muon-catalysed fusion no longer becomes cost-effective.
So the question is will we find a way around any of these problems? As with many things, it depends on human scientific ingenuity, and the practical limits placed upon it. Fusion research is relatively expensive, so investing in it is an exercise in hope. More precisely it is hope that once a sustainable fusion reactor is invented, it will repay the investment many times over.