Quantum Hall effect: from the drude conductivity model to the Chern-Simons field description

The goal of this article is to introduce the quantum Hall effect (QHE) from its classical roots to the quantum integer and fractional versions. This is a robust phenomenon with topological nature. Regarding the fractional quantum Hall effect, its phenomenology and the possibility of vortices with fractional charges are also discussed. The lattice structure of graphene, one of the possible planar matter samples in which quantum Hall effects can occur, is also analyzed. The calculation of the Landau levels occurring in this specific substrate is carefully performed. For completeness, a field description for a whole class of these systems, the ones related to the fractional quantum Hall effect, is also reviewed.