The numerical renormalization group (NRG) method is a powerful tool for studying quantum impurity systems, which consist of a small system (the impurity) coupled to a large system (the environment or bath). The NRG was first developed by Wilson in the 1970s to solve the Kondo problem, which describes the interaction between a magnetic impurity and conduction electrons. The method allows for non-perturbative calculations of the crossover from high-temperature to low-temperature phases in such systems. Over the past 30 years, the NRG has been generalized to a wide range of quantum impurity problems, including the two-channel Kondo model, spin-boson models, and lattice systems within the dynamical mean-field theory (DMFT). The NRG approach involves a logarithmic discretization of the conduction band, followed by a mapping of the discretized model onto a semi-infinite chain. The impurity spin is then represented as the first site of the chain, and the Hamiltonian is iteratively diagonalized. This process allows for the calculation of various physical quantities, including thermodynamic and static properties, as well as dynamic properties such as transport and self-energy. The method has been successfully applied to a variety of systems, including the Kondo effect, quantum dot physics, and impurity quantum phase transitions. The NRG has also been extended to lattice models within DMFT, where the Hubbard model and other correlated electron systems are mapped onto single-impurity Anderson models. This approach has been used to study phenomena such as the Mott transition and the behavior of impurities in superconductors. Additionally, the NRG has been applied to systems with bosonic degrees of freedom, such as the spin-boson model, and to quantum impurities coupled to phonon modes. The NRG method is particularly useful for systems with a broad energy spectrum, as it allows for the efficient calculation of physical quantities in such systems. The method has been shown to provide accurate results for a wide range of quantum impurity problems, including the Kondo effect, two-channel Kondo physics, and impurity quantum phase transitions. The NRG has also been used to study dissipative quantum systems and to calculate dynamic quantities such as spectral functions and transport properties. In summary, the NRG method is a versatile and powerful tool for studying quantum impurity systems. It has been successfully applied to a wide range of physical phenomena, including the Kondo effect, quantum dot physics, and impurity quantum phase transitions. The method has been generalized to lattice models within DMFT and has been used to study a variety of correlated electron systems. The NRG continues to be an important tool in the study of quantum impurity systems, with ongoing research aimed at improving its accuracy and expanding its applicability.The numerical renormalization group (NRG) method is a powerful tool for studying quantum impurity systems, which consist of a small system (the impurity) coupled to a large system (the environment or bath). The NRG was first developed by Wilson in the 1970s to solve the Kondo problem, which describes the interaction between a magnetic impurity and conduction electrons. The method allows for non-perturbative calculations of the crossover from high-temperature to low-temperature phases in such systems. Over the past 30 years, the NRG has been generalized to a wide range of quantum impurity problems, including the two-channel Kondo model, spin-boson models, and lattice systems within the dynamical mean-field theory (DMFT). The NRG approach involves a logarithmic discretization of the conduction band, followed by a mapping of the discretized model onto a semi-infinite chain. The impurity spin is then represented as the first site of the chain, and the Hamiltonian is iteratively diagonalized. This process allows for the calculation of various physical quantities, including thermodynamic and static properties, as well as dynamic properties such as transport and self-energy. The method has been successfully applied to a variety of systems, including the Kondo effect, quantum dot physics, and impurity quantum phase transitions. The NRG has also been extended to lattice models within DMFT, where the Hubbard model and other correlated electron systems are mapped onto single-impurity Anderson models. This approach has been used to study phenomena such as the Mott transition and the behavior of impurities in superconductors. Additionally, the NRG has been applied to systems with bosonic degrees of freedom, such as the spin-boson model, and to quantum impurities coupled to phonon modes. The NRG method is particularly useful for systems with a broad energy spectrum, as it allows for the efficient calculation of physical quantities in such systems. The method has been shown to provide accurate results for a wide range of quantum impurity problems, including the Kondo effect, two-channel Kondo physics, and impurity quantum phase transitions. The NRG has also been used to study dissipative quantum systems and to calculate dynamic quantities such as spectral functions and transport properties. In summary, the NRG method is a versatile and powerful tool for studying quantum impurity systems. It has been successfully applied to a wide range of physical phenomena, including the Kondo effect, quantum dot physics, and impurity quantum phase transitions. The method has been generalized to lattice models within DMFT and has been used to study a variety of correlated electron systems. The NRG continues to be an important tool in the study of quantum impurity systems, with ongoing research aimed at improving its accuracy and expanding its applicability.