Why are insulators insulating and metals conducting?

At variance with what happens in metals, the electronic charge in insulators cannot flow freely under an applied de field, and undergoes instead static polarization. These two features arise from the difference in nature of the excitation spectra, but also from the difference in organization of the electrons in their ground state: electrons are localized in insulators and delocalized in metals. Such localization, however, is hidden in a rather subtle way in the many-body wavefunction. We review the theory of the insulating state, on the basis of electron localization, addressing on the same basis all insulators: either independent electron or correlated, either crystalline or disordered. The starting point is a 1964 milestone paper by Kohn. Significant advances occurred from 1999 onwards. These advances are deeply rooted in the modern theory of polarization: localization and polarization can be regarded as two aspects of the same phenomenon, and stem from essentially the same formalism. Starting from the many-body ground wavefunction, one defines a dimensionless complex number which vanishes in metals and is finite in insulators; in the latter case, its phase (the Berry phase) yields the macroscopic polarization, while its modulus measures localization. Conductivity features are addressed within the same theoretical scheme