  Electronic bands in crystals are described by an ensemble of Bloch wave
functions indexed by momenta defined in the first Brillouin Zone, and their
associated energies. In an insulator, an energy gap around the chemical
potential separates valence bands from conduction bands. The ensemble of
valence bands is then a well defined object, which can possess non-trivial or
twisted topological properties. In the case of a twisted topology, the
insulator is called a topological insulator. We introduce this notion of
topological order in insulators as an obstruction to define the Bloch wave
functions over the whole Brillouin Zone using a single phase convention.
Several simple historical models displaying a topological order in dimension
two are considered. Various expressions of the corresponding topological index
are finally discussed.
