One in a series of talks from the 2019 Models of Consciousness conference. Johannes Kleiner, Institute for Theoretical Physics, Leibniz University of Hannover. The goal of this talk is to discuss the mathematical basis of models of consciousness, most notably the question of which mathematical structure one is to use to describe experience in any formal theory. This question is inextricably linked with the conceptual basis of models of consciousness. After briefly reviewing the various choices made in existing models, I explain how a systematic answer to this question can be constructed based on a further development of the concepts introduced by Thomas Nagel and David Chalmers. If time permits, I will outline how this leads to a full mathematical framework for models of consciousness. Filmed at the Models of Consciousness conference, University of Oxford, September 2019 - https://models-of-consciousness.org/ Creative Commons Attribution-Non-Commercial-Share Alike 2.0 UK: England & Wales; http://creativecommons.org/licenses/by-nc-sa/2.0/uk/