The development of various approaches to describing the complete fusion of nuclei and their connections with experimental studies is discussed. A brief account of the dinuclear-system concept (DNSC), the approach proposed at Dubna, is given. The DNSC revealed two important features of the complete fusion of massive nuclei: the existence of the inner fusion barrier B fus * and the competition between complete fusion and quasifission channels in a dinuclear system formed at the capture stage. The DNSC was applied to the analysis of reactions used to synthesize superheavy elements (SHE). The DNSC provided a basis for the models of competition between complete-fusion and quasifission channels. Using these models, one can describe the cross section for SHE production in cold-and warm-fusion reactions.