  Topological insulators are solid state systems of independent electrons for
which the Fermi level lies in a mobility gap, but the Fermi projection is
nevertheless topologically non-trivial, namely it cannot be deformed into that
of a normal insulator. This non-trivial topology is encoded in adequately
defined invariants and implies the existence of surface states that are not
susceptible to Anderson localization. This non-technical review reports on
recent progress in the understanding of the underlying mathematical structures,
with a particular focus on index theory.
