The self-consistent-charge density-functional tight-binding method (SCC-DFTB) is an approximate quantum chemical method derived from density functional theory (DFT). This study extends and improves SCC-DFTB by incorporating an improved Coulomb interaction between atomic partial charges and a complete third-order expansion of the DFT total energy, leading to the next generation of the DFTB methodology called DFTB3. DFTB3 significantly improves the description of charged systems containing C, H, N, O, and P, especially regarding hydrogen binding energies and proton affinities. It is particularly applicable to biomolecular systems. Challenges and possible solutions are briefly discussed. SCC-DFTB is an approximate approach derived from DFT by neglecting, approximating, and parametrizing interaction integrals. Although less accurate than DFT and ab initio methods, SCC-DFTB offers increased computational speed, making it suitable for large molecules and simulations. The non-self-consistent version of DFTB and its basic integral approximations were proposed in the 1980s and remain central to later extensions. The DFTB energy is a stationary approximation to the DFT functional, similar to other empirical tight-binding models. SCC-DFTB extends DFTB to charge self-consistency and can be derived by a second-order expansion of the DFT total energy with respect to charge density fluctuations. The SCC-DFTB model now allows treatment of systems with intermediate charge transfer within a molecule, making it a major step toward a generally applicable DFT-based semi-empirical methodology. Several reviews have appeared on the basic formalism and selected applications, and a special issue of JPCA is recommended for a recent overview. Recent benchmark studies of SCC-DFTB show its great success and limitations. Geometries are usually reproduced excellently, and relative energies of peptide conformers and hydrogen bonding energies are nicely reproduced. While SCC-DFTB performs well on average for reaction energies, heats of formation are overestimated. Vibrational frequencies are reasonable, but severe failures have been noted for certain vibrational modes. A drawback inherited from DFT is the missing dispersion interaction, which is addressed by an empirical correction. The SCC-DFTB total energy consists of three terms: the DFTB matrix elements, the DFTB repulsive potential, and the second-order term of the DFT Taylor series expansion. The inclusion of approximate third-order terms leads to a new degree of self-consistency. The Coulomb repulsion from charge density fluctuations is computed in a monopole approximation using a newly introduced parameter, the Hubbard parameter (chemical hardness). This parameter is computed from DFT for neutral atoms and is a constant for all charge states of the atom. The third-order terms can be split into diagonal and off-diagonal parts. The diagonal terms lead to a charge-dependent on-site self-interaction, while the off-diagonal terms modify the SCC Coulomb repulsionThe self-consistent-charge density-functional tight-binding method (SCC-DFTB) is an approximate quantum chemical method derived from density functional theory (DFT). This study extends and improves SCC-DFTB by incorporating an improved Coulomb interaction between atomic partial charges and a complete third-order expansion of the DFT total energy, leading to the next generation of the DFTB methodology called DFTB3. DFTB3 significantly improves the description of charged systems containing C, H, N, O, and P, especially regarding hydrogen binding energies and proton affinities. It is particularly applicable to biomolecular systems. Challenges and possible solutions are briefly discussed. SCC-DFTB is an approximate approach derived from DFT by neglecting, approximating, and parametrizing interaction integrals. Although less accurate than DFT and ab initio methods, SCC-DFTB offers increased computational speed, making it suitable for large molecules and simulations. The non-self-consistent version of DFTB and its basic integral approximations were proposed in the 1980s and remain central to later extensions. The DFTB energy is a stationary approximation to the DFT functional, similar to other empirical tight-binding models. SCC-DFTB extends DFTB to charge self-consistency and can be derived by a second-order expansion of the DFT total energy with respect to charge density fluctuations. The SCC-DFTB model now allows treatment of systems with intermediate charge transfer within a molecule, making it a major step toward a generally applicable DFT-based semi-empirical methodology. Several reviews have appeared on the basic formalism and selected applications, and a special issue of JPCA is recommended for a recent overview. Recent benchmark studies of SCC-DFTB show its great success and limitations. Geometries are usually reproduced excellently, and relative energies of peptide conformers and hydrogen bonding energies are nicely reproduced. While SCC-DFTB performs well on average for reaction energies, heats of formation are overestimated. Vibrational frequencies are reasonable, but severe failures have been noted for certain vibrational modes. A drawback inherited from DFT is the missing dispersion interaction, which is addressed by an empirical correction. The SCC-DFTB total energy consists of three terms: the DFTB matrix elements, the DFTB repulsive potential, and the second-order term of the DFT Taylor series expansion. The inclusion of approximate third-order terms leads to a new degree of self-consistency. The Coulomb repulsion from charge density fluctuations is computed in a monopole approximation using a newly introduced parameter, the Hubbard parameter (chemical hardness). This parameter is computed from DFT for neutral atoms and is a constant for all charge states of the atom. The third-order terms can be split into diagonal and off-diagonal parts. The diagonal terms lead to a charge-dependent on-site self-interaction, while the off-diagonal terms modify the SCC Coulomb repulsion