The article reviews the theory and applications of maximally localized Wannier functions (MLWFs), which are localized representations of the electronic ground state of a periodic system. The authors discuss the connection between Bloch and Wannier representations, the gauge freedom in Bloch functions, and the methods for constructing MLWFs through projection or maximal localization. They highlight the importance of the Marzari-Vanderbilt approach, which introduces a localization criterion to identify a unique set of MLWFs for a given crystalline insulator. The review covers various applications of MLWFs, including their use in analyzing chemical bonding, local probes of electronic polarization and orbital magnetization, and their role in constructing model Hamiltonians for correlated electron and magnetic systems. The article also discusses Wannier interpolation schemes and their applications in quantum transport properties, semiempirical potentials, and strongly correlated systems. Finally, it explores the construction and use of MLWFs in contexts beyond electronic structure theory, such as phonons, photonic crystals, and cold atom lattices.The article reviews the theory and applications of maximally localized Wannier functions (MLWFs), which are localized representations of the electronic ground state of a periodic system. The authors discuss the connection between Bloch and Wannier representations, the gauge freedom in Bloch functions, and the methods for constructing MLWFs through projection or maximal localization. They highlight the importance of the Marzari-Vanderbilt approach, which introduces a localization criterion to identify a unique set of MLWFs for a given crystalline insulator. The review covers various applications of MLWFs, including their use in analyzing chemical bonding, local probes of electronic polarization and orbital magnetization, and their role in constructing model Hamiltonians for correlated electron and magnetic systems. The article also discusses Wannier interpolation schemes and their applications in quantum transport properties, semiempirical potentials, and strongly correlated systems. Finally, it explores the construction and use of MLWFs in contexts beyond electronic structure theory, such as phonons, photonic crystals, and cold atom lattices.