Colloquium: Supersolids: What and where are they?

The ongoing experimental and theoretical effort aimed at understanding nonclassical rotational inertia in solid helium has sparked renewed interest in the supersolid phase of matter, its microscopic origin and character, and its experimental detection. The purpose of this Colloquium is to provide a general theoretical framework for the phenomenon of supersolidity, review some of the experimental evidence for solid He4, and discuss its possible interpretation in terms of physical effects underlain by extended defects (such as dislocations). Quantitative support to our theoretical scenarios by means of first-principle numerical simulations is provided. Alternate avenues for the observation of the supersolid phase, not involving helium but rather assemblies of ultracold atoms, are also discussed.