A crystal with its strictly periodic atomic structure represents a natural, very symmetrical three-dimensional diffraction grating for wavelengths of the order of the interatomic distances. Crystal structure analysis is thus based on the theories of symmetry (space groups) on the one hand and of interaction of radiation with solids (diffraction) on the other. The diffraction techniques, with the help of extensive computer calculations, lead to the atomic arrangements of crystalline materials. Depending on the problem, X-rays, electrons, or neutJ:ons are used, which provide the electron-density, electrostatic potential density and nuclear density (also magnetic spin density) distribution, respectively, in a crystal.All possible arrangements of atoms in crystals are governed by the 230 types of space groups, derived in 1890 by the crystallographer E.S. Fedorov and the mathematician A. Schoenflies. X-ray diffraction of crystals was discovered by the physicists M. Von Laue, W. Friedrich, and P. Knipping in 1912. In 1913-1914H. Bragg and W.L. Bragg first applied X-ray diffraction to the experimental confirmation of the structures of single crystals of NaCl, Cu, diamond, etc. previously predicted by W. Barlow. The application of X-ray diffraction was extended to polycrystalline materials in 1916 by P. Debye and P. Sherrer. Details of the history of crystallography can be found in a recent Historical Atlas by 1. Lima-de-Faria.The power of a structure analysis of a synthetic crystal or a mineral and the reliability and accuracy of the results depend on many factors: sample quality, radiation source, apparatus and techniques available, especially for the measurement of the diffracted intensities. The first problem of a structural study is the determination of the symmetry and the lattice parameters of the crystal. The next step is the derivation of an atomic model. The structure is then refined from the diffraction data, taking into account the finer effects of the interaction between sample and radiation. The final stage is the crystallochemical analysis which comprises interpretation of the geometrical model, calculation of interatomic distances, valency angles, sizes, and orientations of thermal-motion ellipsoids and, finally, generation of visible structure models on a display or a plotter.Further advances of the theory and the methods of structure analysis provide information not only on the geometry but also on more subtle features of the mineral structure, such as isomorphous replacement, disorder, variable atomic occupation of crystallographic sites, sizes and misorientation of mosaic blocks in single crystals, as well as thermal motion of the atoms including its anisotropy