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In this paper we describe three lines of research related to the issue of helping robots imitate people. These studies are based on observed human behaviour, technical metrics and implemented technical solutions. The three lines of research are: (a) a number of user studies that show how humans naturally tend to demonstrate a task for a robot to learn, (b) a formal approach to tackle the problem of what a robot should imitate, and (c) a technology-driven conceptual framework and technique, inspired by social learning theories, that addresses how a robot can be taught. In this merging exercise we will try to propose a way through this problem space, towards the design of a Human-Robot Interaction (HRI) system able to be taught by humans via demonstration.