The results of atomistic calculations of long-period tilt boundaries, which were reported in the preceding parts I and II, are generalized and represented concisely by using two-dimensional lattices, called decomposition lattices. The basis vectors of a decomposition lattice characterize the two fundamental structural elements composing all boundaries in a continuous series of boundary structures. Conversely, the governing condition on the basis vectors is that the boundary structure can change continuously throughout the misorientation range between the boundaries represented by the basis vectors. On assuming that no discontinuous changes in boundary structure occur at non-favoured boundary orientations, and that all boundaries considered are stable with respect to faceting, the governing condition may be used to deduce selection rules for adjacent favoured boundaries and the existence of other favoured boundaries in the misorientation range between two given favoured boundaries. The necessary condition for a discontinuous change in boundary structure to be possible at a non-favoured boundary orientation is formulated. Various aspects of intrinsic and extrinsic grain boundary dislocations (g.b.ds) are treated. It is first shown that the observation of intrinsic g.b.d. networks in the transmission electron microscope does not necessarily imply that the reference structure, preserved by those g.b.ds, is a favoured boundary. Secondly, it is argued that extrinsic g.b.ds provide imperfect steps with Burgers vector components parallel to the boundary that do not exist in equilibrium high-angle tilt boundaries. Finally, an explanation of the physical basis of plane matching dislocations is proposed. A general classification of grain boundary properties is introduced that is based on the results of this investigation of grain boundary structure. It is argued that only properties, such as grain boundary diffusion, that depend exclusively on the atomic structure of the boundary core may be used to detect favoured boundaries. Favoured boundaries exist at those misorientations where such a property is continuous but its first derivative, with respect to misorientation, is not. Grain boundary diffusion, the energy against misorientation relation and grain boundary sliding and migration are then discussed.