A review is presented of work related to two new processes which arise in crystalline materials when impurities diffuse along grain boundaries from the surface of the material: diffusion-initiated grain boundary migration and recrystallization. We analyze the conditions under which DIGM occurs, the kinetics of the process, its driving forces, and also the changes in the grain fine structure and the near-grain regions, as well as the nucleation of new grains on the migrating boundaries, caused by an uncompensated impurity atom flux. We consider the mechanisms for DIGM. It is shown that not one of the mechanisms proposed to describe DIGM is capable of explaining all the experimentally observed properties of grain boundary migration under DIGM conditions. We note that changes in the grain boundary structure caused by the impurity atoms diffusing along it are due to diverse grain-boundary processes, which have important technological implications.