In the first part, I will briefly explain the integer quantum Hall effect, starting from its experimental discovery in 1980 by K. von Klitzing, G. Dorda, and M. Pepper, and then reviewing some of the fundamental concepts which shaped its understanding. In the second part, I will present a formula for the longitudinal conductivity for a wide class of two-dimensional tight-binding models, whose Hamiltonian displays conical intersections of the Bloch bands at the Fermi level. Our setting allows to consider generic transitions between quantum Hall phases. The second part is based on a recent joint work with L. Pigozzi and M. Porta: arxiv.org/abs/2503.01381.