In a periodic potential the wave functions of electrons in a solid have the Bloch form and are essentially the modes of a free electron modulated by a periodic amplitude. The problem of free electrons in different dimensions is studied in detail and the notion of crystalline momentum is presented, differing from the usual, free space, momentum, since the continuum translational invariance is transformed into a discrete translational invariance. This involves the possibility to define momenta minus a reciprocal lattice vector. The band theory of electrons is presented, the notion of Brillouin zones is introduced and its consequence for the transport properties is explained. Since electrons are fermions, the statistics plays an important role, and the notion of the existence of a Fermi surface, together with the existence of Brillouin zones, leads to interesting behaviors. Various solids are discussed, from metals to semiconductors, semimetals and insulators. The topological properties of the bands are discussed and concepts such as the Berry phase, winding number and Chern number are introduced and their role is emphasized. The energy bands of graphene are considered as an example of a semimetal.