(Slides: part 1, part 2, part 3, part 4)

We will begin by reviewing the experimental facts defining the Classical and the Quantum Hall Effect (QHE), both integer and fractional. In the simplest setting the integer values of the Hall conductance will be explained in terms of Landau levels, and the role of disorder in the formation of plateaus will be addressed semi-classically. More generally, we will review various characterizations of the Hall conductance (due to Laughlin, Thouless, and Halperin) in terms of a charge pumped by a magnetic flux, of a bulk current induced by an electric field, or of edge currents. Each approach comes with a seemingly different explanation of integrality. In a more mathematical part we will introduce corresponding definitions of the Hall conductance and establish theorems about their equivalence. In a last part we will briefly treat the fractional QHE and possibly the Quantum Spin Hall Effect. It is expected that participants are familiar with basic notions of Quantum Mechanics.