The development of the theory of the classical crystallographic groups is sketched in this chapter, with emphasis on their interrelations and on their role in crystal chemistry and crystal physics. The derivation of the 32 crystal classes around 1830 and of the 14 (Bravais) lattice types in 1850 culminated in the derivation of the 230 space groups in 1891. After the discovery of X-rays in 1895 and X-ray diffraction in 1912, space-group theory could be successfully applied to the methods of crystal-structure determination. The theory of group–subgroup relations between space groups has evolved during the time from Hermann's fundamental paper of 1928 to the publication of this volume. Several applications have emerged from the results. Attention is given to phase transitions, the relations between crystal structures in crystal chemistry and to the correct determination of the symmetry of a crystal structure.