Autonomous open-ended learning (OEL) robots are able to cumulatively acquire new skills and knowledge through direct interaction with the environment, for example re- lying on the guidance of intrinsic motivations and self-generated goals. OEL robots have a high relevance for applications as they can use the autonomously acquired knowledge to accomplish tasks relevant for their human users. OEL robots, however, encounter an important limitation: this may lead to the acqui- sition of knowledge that is not so much relevant to accomplish the users’ tasks. This work analyses a possible solution to this problem that pivots on the novel concept of ‘purpose’. Purposes indicate what the designers and/or users want from the robot. The robot should use internal representations of purposes, called here ‘desires’, to focus its open-ended exploration towards the acquisition of knowledge relevant to accomplish them. This work contributes to develop a computational framework on purpose in two ways. First, it formalises a framework on purpose based on a three-level motivational hierarchy involving: (a) the purposes; (b) the desires, which are domain independent; (c) specific domain dependent state-goals. Second, the work highlights key challenges highlighted by the framework such as: the ‘purpose-desire alignment problem’, the ‘purpose-goal grounding problem’, and the ‘arbitration between desires’.Overall, the approach enables OEL robots to learn in an autonomous way but also to focus on acquiring goals and skills that meet the purposes of the designers and users.