  In 1964 W. Kohn published the milestone paper "Theory of the insulating
state'", according to which insulators and metals differ in their ground state.
Even before the system is excited by any probe, a different organization of the
electrons is present in the ground state and this is the key feature
discriminating between insulators and metals. However, the theory of the
insulating state remained somewhat incomplete until the late 1990s; this review
addresses the recent developments. The many-body ground wavefunction of any
insulator is characterized by means of geometrical concepts (Berry phase,
connection, curvature, Chern number, quantum metric). Among them, it is the
quantum metric which sharply characterizes the insulating state of matter. The
theory deals on a common ground with several kinds of insulators: band
insulators, Mott insulators, Anderson insulators, quantum Hall insulators,
Chern and topological insulators
