Maximally localized Wannier functions (MLWFs) provide a localized representation of electronic states in solids, offering insights into chemical bonding and material properties. Introduced by Gregory Wannier in 1937, Wannier functions are derived from Bloch functions through unitary transformations, enabling the construction of localized orbitals. Since 1997, methods have been developed to iteratively transform extended Bloch orbitals into unique MLWFs, allowing for the analysis of chemical bonding, electric polarization, and orbital magnetization. These functions are also used as efficient basis sets for quantum transport, semi-empirical potentials, and strongly correlated systems. The theory of Wannier functions involves gauge freedom, which introduces non-uniqueness in their definition. This is addressed through projection methods and maximally localized criteria, ensuring that the resulting WFs are well-localized in real space. The localization functional, based on the spread of WFs around their centers, is minimized to achieve maximal localization. This process is computationally feasible and has been implemented in electronic-structure calculations, enabling the construction of MLWFs for various materials. Wannier interpolation schemes allow for the efficient computation of properties on fine k-space meshes using coarse mesh data, enhancing the practicality of Wannier functions. Applications of MLWFs extend beyond electronic-structure theory to phonon excitations, photonic crystals, and cold-atom optical lattices. The use of WFs as basis functions in these contexts has led to significant advancements in understanding and modeling complex systems. Overall, MLWFs provide a powerful tool for analyzing and predicting material properties, bridging the gap between electronic structure theory and experimental observations.Maximally localized Wannier functions (MLWFs) provide a localized representation of electronic states in solids, offering insights into chemical bonding and material properties. Introduced by Gregory Wannier in 1937, Wannier functions are derived from Bloch functions through unitary transformations, enabling the construction of localized orbitals. Since 1997, methods have been developed to iteratively transform extended Bloch orbitals into unique MLWFs, allowing for the analysis of chemical bonding, electric polarization, and orbital magnetization. These functions are also used as efficient basis sets for quantum transport, semi-empirical potentials, and strongly correlated systems. The theory of Wannier functions involves gauge freedom, which introduces non-uniqueness in their definition. This is addressed through projection methods and maximally localized criteria, ensuring that the resulting WFs are well-localized in real space. The localization functional, based on the spread of WFs around their centers, is minimized to achieve maximal localization. This process is computationally feasible and has been implemented in electronic-structure calculations, enabling the construction of MLWFs for various materials. Wannier interpolation schemes allow for the efficient computation of properties on fine k-space meshes using coarse mesh data, enhancing the practicality of Wannier functions. Applications of MLWFs extend beyond electronic-structure theory to phonon excitations, photonic crystals, and cold-atom optical lattices. The use of WFs as basis functions in these contexts has led to significant advancements in understanding and modeling complex systems. Overall, MLWFs provide a powerful tool for analyzing and predicting material properties, bridging the gap between electronic structure theory and experimental observations.