Social intelligence in robots has a quite recent history in artificial intelligence and robotics. However, it has become increasingly apparent that social and interactive skills are necessary requirements in many application areas and contexts where robots need to interact and collaborate with other robots or humans. Research on human–robot interaction (HRI) poses many challenges regarding the nature of interactivity and ‘social behaviour’ in robot and humans. The first part of this paper addresses dimensions of HRI, discussing requirements on social skills for robots and introducing the conceptual space of HRI studies. In order to illustrate these concepts, two examples of HRI research are presented. First, research is surveyed which investigates the development of a cognitive robot companion. The aim of this work is to develop social rules for robot behaviour (a ‘robotiquette’) that is comfortable and acceptable to humans. Second, robots are discussed as possible educational or therapeutic toys for children with autism. The concept of interactive emergence in human–child interactions is highlighted. Different types of play among children are discussed in the light of their potential investigation in human–robot experiments. The paper concludes by examining different paradigms regarding ‘social relationships’ of robots and people interacting with them.