The meaning of “social dominance” and the problems associated with its use are discussed. Most problems disappear if dominance is restricted to the phenomenon that in every pair of animals one member can inhibit the behaviour of the other. The dominance order of the group is the sum of all such inhibitory relationships. Dominant animals probably have been aggressive in the past to obtain their dominant positions, but a dominant animal need not be aggressive now. Measures of the dominance position of animals in a herd should be based on observations in the particular herd, contain sufficient observations to be reliable, reflect the actual magnitude of differences between animals, and be normally distributed. The method of Beilharz and Mylrea (1963) meets these requirements. The application of this method is discussed. The dominance structure of three dairy herds near Freiburg in South West Germany were studied. None of the herds had an individual cow who was dominant over all others in the herd. Parameters of social structure varied from herd to herd. The findings are discussed in relation to differences in space per animal, manner of recruitment of young stock to the herds, presence of horns and other features. It is concluded that the dominance relationship of any pair of animals is a result of learning, with many different factors, including trivial ones, being involved in the initial serious formation of the relationship. Once learnt, dominance relationships for each pair of animals persist for a long time. While bidirectional dominance relationships are found in young animals, where none has yet accepted dominance of others, mature animals of all species of domestic livestock generally have clear unidirectional dominance relationships. The dominance order of the group may be no more than the sum of the individual relationships. Dominant animals are freer than others in attending to stimuli from outside the herd.