In this chapter I discuss the problem of electrons moving on a plane in the presence of an external uniform magnetic field perpendicular to the system. This is a subject of great interest from the point of view of both theory and experiment. The explanation of the remarkable quantization of the Hall conductance observed in MOSFETs and in heterostructures has demanded a great deal of theoretical sophistication. Concepts drawn from branches of mathematics, such as topology and differential geometry, have become essential to the understanding of this phenomenon. In this chapter I will consider only the quantum Hall effect in non-interacting systems. This is the theory of the integer Hall effect. The fractional quantum Hall effect (FQHE) is discussed in Chapter 13. The related subject of topological insulators is discussed in Chapter 16.

The chapter begins with a description of the one-electron states, both in the continuum and on a 2D lattice, followed by a summary of the observed phenomenology of the quantum Hall effect. A brief discussion of linear-response theory is also presented. The rest of the chapter is devoted to the problem of topological quantization of the Hall conductance.