In this talk I will give a pedagogical introduction to the quantum mechanical description of extended systems. I will first introduce the concept of band structure and its link to the electronic structure of finite systems. I will then move to the most commonly used ab-initio theories to describe some interesting properties (e.g. photoemission and optical spectra) of these systems. I will focus in particular on density-functional theory [1] (and its extension to time) and on many-body perturbation theory based on Green's functions [2].

[1] P. Hohenberg, and W. Kohn 136. B864 (1964).
[2] A. Fetter and J. D. Walecka. Quantum Theory of Many-Particle Systems. Dover publications, 2003; L. Hedin, Phys. Rev. 139, A796 (1965).