The way human beings engage with material things in our environment is experiencing rapid modification. Human and non-human, natural and artificial creatures are on the verge of building unprecedented relations of sociability. This paper takes this process as a horizon for Social Robotics, advancing a new approach to coordinate systems of multiple robots within social spaces durably shared by humans and machines. Given the fact that institutions are the tools in use within human societies to shape social action over long periods of time, we use human-inspired institutions to deal with scenarios involving many-to-many human-robot lasting interactions. Our approach, Institutional Robotics, is inspired by leading economists and philosophers having dedicated sustained efforts to the understanding of social institutions. This paper: (1) advocates the importance of an institution-based approach for multi-robot systems (Institutional Robotics) in real-world human-populated environments, where many-to-many social interactions among robots and humans must be considered; (2) reviews experiments conducted (including novel experimental work) and methodologies used in the process of advancing Institutional Robotics. Both contributions pave the way for a new institution-based methodology to coordinate robot collectives, which stems from an inter-disciplinary approach based on robotics, social sciences and philosophy.