Ensuring the trustworthiness of autonomous systems and artificial intelligence is an important interdisciplinary endeavour. In this position paper, we argue that this endeavour will benefit from technical advancements in capturing various forms of responsibility, and we present a comprehensive research agenda to achieve this. In particular, we argue that ensuring the reliability of autonomous system can take advantage of technical approaches for quantifying degrees of responsibility and for coordinating tasks based on that. Moreover, we deem that, in certifying the legality of an AI system, formal and computationally implementable notions of responsibility, blame, accountability, and liability are applicable for addressing potential responsibility gaps (i.e. situations in which a group is responsible, but individuals’ responsibility may be unclear). This is a call to enable AI systems themselves, as well as those involved in the design, monitoring, and governance of AI systems, to represent and reason about who can be seen as responsible in prospect (e.g. for completing a task in future) and who can be seen as responsible retrospectively (e.g. for a failure that has already occurred). To that end, in this work, we show that across all stages of the design, development, and deployment of trustworthy autonomous systems (TAS), responsibility reasoning should play a key role. This position paper is the first step towards establishing a road map and research agenda on how the notion of responsibility can provide novel solution concepts for ensuring the reliability and legality of TAS and, as a result, enables an effective embedding of AI technologies into society.