A digital twin is a virtual data-driven replica of a real-world system. Recently, digital twins have become popular in engineering and infrastructure planning where they provide insights into complex physical systems or processes. Yet, to date, considerably less research has explored how digital replicas of social systems – representing the decisions, behaviors and interactions of individual people, and, in turn, their emergent outcomes – might be developed and integrated with those of physical systems. In this position paper we discuss the need for such societal twins, what they might look like, and set out key challenges that will need to be overcome if these benefits are to be realised.