The quantum Hall effect is the precise quantization, in units e2/h, of the Hall conductance of a system of charge carriers constrained to move in a plane, with a perpendicular magnetic field. It was discovered experimentally by von Klitzing et al. in 1980, and successful theories were put forward soon after. One approach to the theory involves concepts of algebraic topology; the Hall conductance is expressed as a topological invariant called the Chern class. This article is an informal review of the topological theory of the integer quantum Hall effect, including a non-technical introduction to the mathematical concepts involved.