The quantum Hall effect (QHE) refers to the fact that the Hall conductance of a two-dimensional electron gas takes on only quantised values, i.e. integer (or fractional) multiples of e^2/h. First discovered in 1980, it has sparked significant experimental, theoretical and mathematical research, with Nobel Prizes being awarded to both experimentalists and theorists. In this series of blackboard lectures, I will discuss recent developments that demonstrate how certain experimentally relevant features of the QHE can be rigorously understood within microscopic models of interacting fermions. Most importantly, I will explain results that establish a rigorous Ohm’s law for macroscopic Hall currents in interacting, spectrally gapped systems at zero temperature. Familiarity with the fundamental mathematical principles of quantum mechanics is expected. However, I will introduce the relevant operator algebraic framework during the lectures.