The concept of dominance has contributed greatly to our understanding of social structure in animals. Over the past three decades, however, a variety of concepts and definitions of dominance have been introduced, leading to an ongoing debate about the usefulness and meaning of the concept. Criticisms aimed at one definition of dominance do not necessarilly apply to other definitions. Existing definitions can be structural or functional, refer to roles or to agonistic behaviour, regard dominance as a property of individuals or as an attribute of dyadic encounters, concentrate on aggression or on the lack of it, and be based either on theoretical constructs or on observable behaviour. Thirteen definitions of dominance are reviewed, and their usefulness assessed with respect to their descriptive value. The predictive and explanatory values of definitions are specific to the questions asked in each particular study and are not considered as criteria to judge the usefulness of the dominance concept. By virtue of its high descriptive value, the original definition of dominance by SCHJELDERUPP-EBBE (1922, Z.Psychol. 88: 226-252) emerged as the basis to formulate a structural definition with wide applicability and which reflects the essence of the concept: Dominance is an attribute of the pattern of repeated, agonistic interactions between two individuals, characterized by a consistent outcome in favour of the same dyad member and a default yielding response of its opponent rather than escalation. The status of the consistent winner is dominant and that of the loser subordinate. Dominance status refers to dyads while dominance rank, high or low, refers to the position in a hierarchy and, thus, depends on group composition. Dominance is a relative measure and not an absolute property of individuals. The discussion includes reference to the heritability of dominance, application of dominance to groups rather than individuals, and the role of individual recognition and memory during agonistic encounters.