This review presents a family of atomistic, mostly quantum chemistry (QC) based semiempirical methods for the fast and reasonably accurate description of large molecules in gas and condensed phases. These methods, known as extended tight-binding (xTB), are derived from a density functional (DFT) perturbation expansion of the electron density in fluctuation terms, similar to the original DFTB model. The term "extended" emphasizes the availability of parameters for almost the entire periodic table (Z ≤ 86) and improvements in the underlying theory, including atomic orbital basis sets, multipole approximations, and treatment of electrostatic and dispersion interactions. A common feature is consistent parameterization on accurate gas-phase reference data for geometries, vibrational frequencies, and noncovalent interactions. Specialized versions are used for electronic spectra and response properties. The review discusses various benchmarks for structural and thermochemical properties, including transition-metal systems. Recent extensions include the force-field (FF) level and applications to solids under periodic boundary conditions. The xTB family is attractive due to its excellent cost-accuracy ratio and robustness, making it suitable for various fields of computer-aided chemical research. The review covers the theory, implementation details, and applications of xTB methods, including GFN1-xTB, GFN2-xTB, and GFN0-xTB. These methods are based on a Taylor expansion of the total energy around a reference density, with different orders of expansion corresponding to different levels of sophistication. The GFN1-xTB method includes terms for repulsion, dispersion, and electrostatic interactions, while GFN2-xTB incorporates anisotropic electrostatic and XC terms, as well as D4 dispersion interactions. The review also discusses the application of xTB methods to solids and the use of periodic boundary conditions. The methods are implemented in the xtb program and are used for a wide range of applications, including molecular structures, thermochemistry, transition states, and proteins. The review highlights the advantages of xTB methods, including their efficiency, accuracy, and robustness, making them a valuable tool in computational chemistry.This review presents a family of atomistic, mostly quantum chemistry (QC) based semiempirical methods for the fast and reasonably accurate description of large molecules in gas and condensed phases. These methods, known as extended tight-binding (xTB), are derived from a density functional (DFT) perturbation expansion of the electron density in fluctuation terms, similar to the original DFTB model. The term "extended" emphasizes the availability of parameters for almost the entire periodic table (Z ≤ 86) and improvements in the underlying theory, including atomic orbital basis sets, multipole approximations, and treatment of electrostatic and dispersion interactions. A common feature is consistent parameterization on accurate gas-phase reference data for geometries, vibrational frequencies, and noncovalent interactions. Specialized versions are used for electronic spectra and response properties. The review discusses various benchmarks for structural and thermochemical properties, including transition-metal systems. Recent extensions include the force-field (FF) level and applications to solids under periodic boundary conditions. The xTB family is attractive due to its excellent cost-accuracy ratio and robustness, making it suitable for various fields of computer-aided chemical research. The review covers the theory, implementation details, and applications of xTB methods, including GFN1-xTB, GFN2-xTB, and GFN0-xTB. These methods are based on a Taylor expansion of the total energy around a reference density, with different orders of expansion corresponding to different levels of sophistication. The GFN1-xTB method includes terms for repulsion, dispersion, and electrostatic interactions, while GFN2-xTB incorporates anisotropic electrostatic and XC terms, as well as D4 dispersion interactions. The review also discusses the application of xTB methods to solids and the use of periodic boundary conditions. The methods are implemented in the xtb program and are used for a wide range of applications, including molecular structures, thermochemistry, transition states, and proteins. The review highlights the advantages of xTB methods, including their efficiency, accuracy, and robustness, making them a valuable tool in computational chemistry.