The density-matrix renormalization group (DMRG) is a numerical algorithm for efficiently truncating the Hilbert space of low-dimensional strongly correlated quantum systems. It has become the method of choice for studying one-dimensional quantum systems due to its high precision and versatility. DMRG is applied to calculate static, dynamic, and thermodynamic quantities in such systems. Its potential is also explored in two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, and time-dependent phenomena. The review discusses the theoretical foundations of DMRG, its relationship to matrix-product states, and its connection to quantum information theory. DMRG is based on real-space renormalization group methods, which involve iteratively integrating out degrees of freedom and modifying the Hamiltonian. The algorithm uses density matrices to truncate the Hilbert space, retaining the most significant states. It is particularly effective for one-dimensional systems but has been extended to higher dimensions and other fields. The method is known for its ability to handle large systems at high precision and its absence of the negative sign problem that limits quantum Monte Carlo methods. The review covers key aspects of DMRG, including its application to static and dynamic properties, the use of matrix-product states, and its role in quantum information theory. It also discusses the infinite-system and finite-system DMRG approaches, the importance of symmetries and good quantum numbers, and the challenges in handling strongly correlated systems. The review highlights the algorithm's effectiveness in studying ground states, excitations, and correlations, as well as its application to various physical systems, including spin chains, quantum chemistry, and nuclear physics. The review concludes with an outlook on the future of DMRG and its potential in advancing our understanding of strongly correlated quantum systems.The density-matrix renormalization group (DMRG) is a numerical algorithm for efficiently truncating the Hilbert space of low-dimensional strongly correlated quantum systems. It has become the method of choice for studying one-dimensional quantum systems due to its high precision and versatility. DMRG is applied to calculate static, dynamic, and thermodynamic quantities in such systems. Its potential is also explored in two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, and time-dependent phenomena. The review discusses the theoretical foundations of DMRG, its relationship to matrix-product states, and its connection to quantum information theory. DMRG is based on real-space renormalization group methods, which involve iteratively integrating out degrees of freedom and modifying the Hamiltonian. The algorithm uses density matrices to truncate the Hilbert space, retaining the most significant states. It is particularly effective for one-dimensional systems but has been extended to higher dimensions and other fields. The method is known for its ability to handle large systems at high precision and its absence of the negative sign problem that limits quantum Monte Carlo methods. The review covers key aspects of DMRG, including its application to static and dynamic properties, the use of matrix-product states, and its role in quantum information theory. It also discusses the infinite-system and finite-system DMRG approaches, the importance of symmetries and good quantum numbers, and the challenges in handling strongly correlated systems. The review highlights the algorithm's effectiveness in studying ground states, excitations, and correlations, as well as its application to various physical systems, including spin chains, quantum chemistry, and nuclear physics. The review concludes with an outlook on the future of DMRG and its potential in advancing our understanding of strongly correlated quantum systems.