The possibility of a supersolid state of matter, i.e., a crystalline solid exhibiting superfluid properties, first appeared in theoretical studies about forty years ago. After a long period of little interest due to the lack of experimental evidence, it has attracted strong experimental and theoretical attention in the last few years since Kim and Chan (Penn State, U.S.A.) reported evidence for nonclassical rotational inertia effects, a typical signature of superfluidity, in samples of solid 4 He. Since this ''first observation'', other experimental groups have observed such effects in the response to the rotation of samples of crystalline helium, and it has become clear that the response of the solid is extremely sensitive to growth conditions, annealing processes, and 3 He impurities. A peak in the specific heat in the same range of temperatures has been reported as well as anomalies in the elastic behaviour of solid 4 He with a strong resemblance to the phenomena revealed by torsional oscillator experiments. Very recently, the observation of unusual mass transport in hcp solid 4 He has also been reported, suggesting superflow. From the theoretical point of view, powerful simulation methods have been used to study solid 4 He, but the interpretation of the data is still rather difficult; dealing with the question of supersolidity means that one has to face not only the problem of the coexistence of quantum coherence phenomena and crystalline order, exploring the realm of spontaneous symmetry breaking and quantum field theory, but also the problem of the role of disorder, i.e., how defects, such as vacancies, impurities, dislocations, and grain boundaries, participate in the phase transition mechanism.