This review covers a family of atomistic, semiempirical quantum chemistry (QC) methods, known as xTB, designed for the fast and reasonably accurate description of large molecules in both gas and condensed phases. The methods are derived from a density functional (DFT) perturbation expansion of the electron density, similar to the original density functional tight binding model. The term "eXtended" emphasizes the wide applicability of these methods to almost all elements (Z ≤ 86) and improvements in the underlying theory, such as the atomic orbital basis set, multipole approximation, and treatment of electrostatic and dispersion interactions. The xTB methods are parameterized consistently on accurate gas phase theoretical reference data for geometries, vibrational frequencies, and noncovalent interactions, which are the primary properties of interest in typical applications to systems composed of up to a few thousand atoms. Specialized versions have been developed for the description of electronic spectra and corresponding response properties. The review includes a theoretical background, implementation details in the efficient and free xTB program, and various benchmarks for structural and thermochemical properties, including (transition-)metal systems. Recent extensions of the model to the force-field (FF) level and its application to solids under periodic boundary conditions are also discussed. The general applicability, excellent cost-accuracy ratio, and high robustness make the xTB family of methods highly attractive for various fields of computer-aided chemical research.This review covers a family of atomistic, semiempirical quantum chemistry (QC) methods, known as xTB, designed for the fast and reasonably accurate description of large molecules in both gas and condensed phases. The methods are derived from a density functional (DFT) perturbation expansion of the electron density, similar to the original density functional tight binding model. The term "eXtended" emphasizes the wide applicability of these methods to almost all elements (Z ≤ 86) and improvements in the underlying theory, such as the atomic orbital basis set, multipole approximation, and treatment of electrostatic and dispersion interactions. The xTB methods are parameterized consistently on accurate gas phase theoretical reference data for geometries, vibrational frequencies, and noncovalent interactions, which are the primary properties of interest in typical applications to systems composed of up to a few thousand atoms. Specialized versions have been developed for the description of electronic spectra and corresponding response properties. The review includes a theoretical background, implementation details in the efficient and free xTB program, and various benchmarks for structural and thermochemical properties, including (transition-)metal systems. Recent extensions of the model to the force-field (FF) level and its application to solids under periodic boundary conditions are also discussed. The general applicability, excellent cost-accuracy ratio, and high robustness make the xTB family of methods highly attractive for various fields of computer-aided chemical research.