A historical introduction outlines the development of experimental and theoretical interest in the diffuse diffraction of X-rays by single crystals. The present experimental study, illustrated by numerous photographs, has consisted in taking Laue, oscillation and rotation photographs, in many different orientations, of inorganic and organic crystals belonging to various systems, using Cu, Mo and Ag radiations, unfiltered, filtered or monochromatized. Rules are laid down for the indexing of diffuse spots and for the determination of the conditions under which such spots appear. Attention is drawn to the presence of non-radial streaks and to the difference in origin of these and of the more usual radial streaks. The relations of the intensity, shape, size, position and persistence of the diffuse spots to the nature, structure, perfection, orientation and temperature of the crystal examined, to the radiation used and to the conditions of photography, are considered in some detail. The use of diffuse spot patterns as a subsidiary method of crystal structure determination is emphasized. It is pointed out that in reciprocal space each reflecting lattice point is surrounded by a region of diffuse scattering, whose physical significance is open to various theoretical interpretations. These diffusely reflecting regions are not, in general, spherical, but are extended along reciprocal lattice planes and axes.