AbstractIn this paper, we review the foundations of the density‐functional tight‐binding (DFTB) method. The method is based on the density‐functional theory as formulated by Hohenberg and Kohn. It introduces several approximations: First, densities and potentials are written as superpositions of atomic densities and potentials. Second, many‐center terms are summarized together with nuclear repulsion energy terms in a way that they can be written as a sum of pairwise repulsive terms. For small distances, the nuclear repulsion dominates, whereas for large distances, these terms vanish. The Kohn–Sham orbitals are expanded in a set of localized atom‐centered functions. They are represented in a minimal basis of optimized atomic orbitals, which are obtained for spherical symmetric spin‐unpolarized neutral atoms self‐consistently. The whole Hamilton and overlap matrices contain one‐ and two‐center contributions only. Therefore, they can be calculated and tabulated in advance as functions of the distance between atomic pairs. In addition, we discuss a self‐consistent charge extension, the treatment of weak interactions, and a linear response scheme in connection with the DFTB method. Finally, some practical aspects are presented. © 2012 John Wiley & Sons, Ltd.This article is categorized under: Electronic Structure Theory > Semiempirical Electronic Structure Methods