It is often convenient or useful in mathematics to treat isomorphic structures as the same. The Univalence Axiom for the foundations of mathematics elevates this idea to a foundational principle in the setting of Homotopy Type Theory. It states, roughly, that isomorphic structures can be identified. In his talk, Prof. Awodey will explain this principle and how it can be taken as an axiom, and explore the motivations and consequences, both mathematical and philosophical, of making such an assumption.