Supersolidity was proposed a long time ago by Andreev and Lifshitz in a Russian paper in 1969, which unfortunately I could never put my hands on, and by Chester and Legget (Nobel prize winner of 2003 exactly by his works on superfluidity and superconductinity) in two different papers in 1970:
 Andreev and Lifshitz, Sov. Phys. JTEP 29, 1107 (1969)
 Chester, Speculations on Bose-Einstein Condensation and Quant um Crystals, Phys. Rev. A 2, 256 (1970)
 Legget, Can a Solid be "Superfluid"?, Phys. Rev. Lett. 25, 1543 (1970)
Actually, Chester does not cite Andreev and Lifshitz's work, which is considered the first one with the idea, but Legget explicitly cites Chester's work. As I said, I don't have access to the Russian work, but I could read the other two. They are very short papers and easily readable.
The idea of a supersolid is a very interesting one that is derived from the phenomenon of superfluidity. Superfluidity is a very low temperature state of a fluid, the most common being Helium, where the viscosity disappears entirely. If a superfluid is put inside a rotating vessel, as long as the rotational velocity stays below a critical velocity, it will stay at rest in the laboratory frame and will not rotate as there will be no friction between the fluid and the walls of the container. Superfluidity is due to a Bose-Einstein condensation of the atoms in the fluid. They all condensate in the same quantum state creating all the interesting effects that you can see partially in this video:
Superfluidity is also related to superconductivity, with the difference that superfluidity is a phenomenon that occur for bosons, while superconductivity occurs for the "electron fluid" that lives in every metal. The idea of supersolidity expressed by Chester was based on the form of the wave-function for the system which he speculated could support both crystalline order and superfluidity, with the most probable system being Helium-4 (2 protons and 2 neutrons). In the end of the paper he speculates that disorder most play a crucial hole with vacancies being the key to the superfluid properties. Vacancies are places in the crystal lattice which were supposed to be occupied by an atom but in fact are empty. It is quite interesting that at this early point disorder was already proposed to play a major role, something that recent experiments seem to corroborate.
Then, in its 1960 paper, Legget gives a more detailed description of the supersolid, although he still doesn't use this name. In the paper he describes a superfluid solid as one with a fraction of its mass condensed in the form of a superfluid and proposes to use what he called the non-classical rotational inertia (NCRI) effect to test it. NCRI is a direct result of the zero viscosity in a superfluid. If you enclose the solid in an annulus and rotate it, the superfluid fraction by its own superfluid nature is expected not to rotate with the container. This results in a decrease in the moment of inertia of the solid, which gives the name to the effect.
In 2004, Kim and Chan used the basic idea of Legget to test for a NCRI in a torsional oscillator (TO). A TO is an annulus with the wannabe supersolid inside that oscillates with some natural resonant frequency that is basically inversely proportional to the square-root of the moment of inertia of the annulus when the damping is small. If the material inside the TO is a supersolid, the experiment is supposed to measure a higher oscillating frequency than for the normal material as moment of inertia should be lower. This change in the resonant frequency was claimed to be measure by Kim and Chan in bulk Helium-4 in two papers, Nature and Science (everybody's dream, I know...):
 Kim and Chan, Probable observation of a supersolid helium phase, Nature 427, 225 (2004).
 Kim and Chan, Observation of Superflow in Solid Helium, Science 305, 1941 (2004).
I did not have access to these papers, so everything I am saying about them are based on what is written in the referencing papers. Kim and Chan observed a possible transition to a supersolid state at a temperature of about 200 mK. These observations were followed by a huge number of other experiments with He-4 with some conflicting results. Although there was some doubt in the beginning if the measured NCRI was really due to supersolidity, it seems now that other explanations are not very probable. These subsequent experiments also seem to corroborate Chester's intuition that the higher the disorder in the sample, the higher the supersolid signal. Usually, disorder appears in the form of vacancies, which was Chester's original idea, or dislocations, which are a kind of defect where the there is some mismatch in the form of the crystalline lattice. A dislocation induces a mismatched growth of the crystal around it. This picture is an example of a screw (for obvious reasons) dislocation in a silicon carbide crystal
and was taken from an article about supersolidity from 2007 named Supersolid, with a Twist commenting a paper publish in Physical Review Letters. I have noticed that many blogs in the internet promote the above picture as being of a supersolid just because it appeared in the above article. THIS IS NOT A SUPERSOLID AND CERTAINLY NOT HELIUM. It's a regular crystal with dislocations. The article seems to indicate that dislocations instead of vacancies are involved in the supersolidity of He-4 by forming a kind of pathway where atoms can move as a superfluid. Actually, experiments have observed that the dependence of the stiffening in solid helium on the macroscopic parameters (temperature, pressure) has the same shape as the dependence of the NCRI. Stiffening (the effect of a solid becoming "harder" in some sense) is related to the mobility of the dislocations in the crystal. These defects can move around the crystal and the more mobile they are, the more malleable is the solid. When the dislocations are somehow pinned to their places, what can occur in the presence of impurities to which they attach, the solid becomes stiffer. Therefore, the relation between stiffness and NCRI indicates that dislocations instead of vacancies are the important kind of defects involved in the onset of supersolidity.
I am writing all this stuff about supersolidity because of a recent article about this to which I have already posted a link to (see Physics - Spotlight of Exceptional Research)
 Sébastien Balibar, Is there a true supersolid phase transition?, Physics 3, 39 (2010)
where the author comments on the recent Physical Review Letters paper
 Oleksandr Syshchenko, James Day, and John Beamish, Frequency Dependence and Dissipation in the Dynamics of Solid Helium, Phys. Rev. Lett. 104, 195301 (2010)
The paper is temporarily free for download, so if you run you can still get it at no charge. What Syshchenko et al. have measured was the dependency of the stiffness with temperature for a range of different oscillation frequencies. That is not possible with a TO technique as it relies on taking measurements exactly at the resonant frequencies of the oscillator. In the experiment, Syshchenko et al. used a sample of solid helium inside a container with one of the surfaces a piezoelectric material that would apply a strain in the sample according to an electric current applied to it.
The discussion by Balibar is a good summary of the work and has also some opinions about the result of the latter. It seems that the experiment indicates that what was thought to be a phase transition to a supersolid state at 200 mK may not be, being instead what they call a crossover. I don't think it is completely clear to me what it means in this case. Usually a crossover is a change of universal exponents, technical quantities that characterise phase transitions, from one universality class to another. In other places I have seen crossover used as a synonymous of a second order phase transitions, something that cannot be the case here as the authors says explicitly that their crossover is not a phase transition, what I think should include second order ones, although I may be wrong. I confess I am a bit confused here and if someone can clarify the issue I would be glad.
In any case, they argued that the true phase transition, according to their definition, must be located somewhere below 55 mK and not around 200 mK. I suppose it must mean that there is no divergence on the susceptibility and probably that's also related to the failure of fitting power laws close to the 200 mK temperature actually. However, technically, this also may be called a phase transition, but of a higher order...
Supersolidity is still a very ill understood phenomena and there is not a good theoretical explanation that can account for all the observed experiments. Each time a new experimental piece of this puzzle appears it shows that our theoretical understanding is still poor. I guess it is a bit tacky to say that as it was already said many times, but I guess it is great to see that there is still so many things that we do not understand.