The theory of grain boundary structure is reviewed from the mathematical point of view. High-order twins are mathematically introduced and developed up to the 4th order twins in the case of cubic crystals. The coincidence-site lattice, 0-lattice and DSC lattice are considered with emphasis on their algebraic aspects. Some applications of high-order twins and coincidence-site boundaries are discussed to point out these importance in the field of material sciences and industries.
