This paper introduces a Theme Issue presenting the latest developments in research on the interplay between flexibility and constraint in social behaviour, using comparative datasets, long-term field studies and experimental data from both field and laboratory studies of mammals. We first explain our focus on mammals and outline the main components of their social systems, focusing on variation within- and among-species in numerous aspects of social organization, mating system and social structure. We then review the current state of primarily ultimate explanations of this diversity in social behaviour. We approach the question of how and why the balance between behavioural flexibility and continuity is achieved by discussing the genetic, developmental, ecological and social constraints on hypothetically unlimited behavioural flexibility. We introduce the other contributions to this Theme Issue against this background and conclude that constraints are often crucial to the evolution and expression of behavioural flexibility. In exploring these issues, the enduring relevance of Tinbergen's seminal paper ‘On aims and methods in ethology’, with its advocacy of an integrative, four-pronged approach to studying behaviour becomes apparent: an exceptionally fitting tribute on the 50th anniversary of its publication.