The so-called D4 model is presented for the accurate computation of London dispersion interactions in density functional theory approximations (DFT-D4) and generally for atomistic modeling methods. In this successor to the DFT-D3 model, the atomic coordination-dependent dipole polarizabilities are scaled based on atomic partial charges which can be taken from various sources. For this purpose, a new charge-dependent parameter-economic scaling function is designed. Classical charges are obtained from an atomic electronegativity equilibration procedure for which efficient analytical derivatives with respect to nuclear positions are developed. A numerical Casimir-Polder integration of the atom-in-molecule dynamic polarizabilities then yields charge- and geometry-dependent dipole-dipole dispersion coefficients. Similar to the D3 model, the dynamic polarizabilities are precomputed by time-dependent DFT and all elements up to radon (Z = 86) are covered. The two-body dispersion energy expression has the usual sum-over-atom-pairs form and includes dipole-dipole as well as dipole-quadrupole interactions. For a benchmark set of 1225 molecular dipole-dipole dispersion coefficients, the D4 model achieves an unprecedented accuracy with a mean relative deviation of 3.8% compared to 4.7% for D3. In addition to the two-body part, three-body effects are described by an Axilrod-Teller-Muto term. A common many-body dispersion expansion was extensively tested, and an energy correction based on D4 polarizabilities is found to be advantageous for larger systems. Becke-Johnson-type damping parameters for DFT-D4 are determined for more than 60 common density functionals. For various standard energy benchmark sets, DFT-D4 slightly but consistently outperforms DFT-D3. Especially for metal containing systems, the introduced charge dependence of the dispersion coefficients improves thermochemical properties. We suggest (DFT-)D4 as a physically improved and more sophisticated dispersion model in place of DFT-D3 for DFT calculations as well as other low-cost approaches like semi-empirical models.