Introduction to the theory of the integer quantum Hall effect

We review the various theoretical arguments which strongly suggest that integer quantization of the Hall conductance holds exactly for a non-relativistic two-dimensional electron system in the presence of a static disorder and electron-electron interactions, when the chemical potential lies in a mobility gap of the system. The semi-classical picture, the Laughlin gauge-invariance argument and the connection between the Kubo Hall conductance and Chern numbers are presented, and the relevance of these idealized geometries to real systems is discussed. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.