One in a series of talks from the 2019 Models of Consciousness conference. Sean Tull Department of Computer Science, University of Oxford Integrated Information Theory (IIT), developed by Giulio Tononi and collaborators, has emerged as one of the leading scientific theories of consciousness. At the heart of IIT is an algorithm which, based on the level of integration of the internal causal relationships of a physical system in a given state, claims to determine the intensity and quality of its conscious experience. However, IIT is known to possess several technical problems, and is only applicable to simple classical physical systems. To be treated as fundamental, it should ideally be extended to more general physical theories. In this work, we investigate the formal structure of IIT, and define a notion of generalised integrated information theory in order to address these problems. Formally such a theory specifies a mapping from a given theory of physics to one of conscious experience, each satisfying minimal conditions needed for the IIT algorithm. In particular we show how a generalisation of IIT may be constructed from any suitable physical process theory, as described mathematically by a symmetric monoidal category. Specialising to classical processes yields IIT as usually defined, while restricting to quantum processes yields the recently proposed Quantum IIT of Zanardi et al. as a special case. Filmed at the Models of Consciousness conference, University of Oxford, September 2019. Creative Commons Attribution-Non-Commercial-Share Alike 2.0 UK: England & Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/