  Many introductions to homotopy type theory and the univalence axiom gloss
over the semantics of this new formal system in traditional set-based
foundations. This expository article, written as lecture notes to accompany a
3-part mini course delivered at the Logic and Higher Structures workshop at
CIRM-Luminy, attempts to survey the state of the art, first presenting
Voevodsky's simplicial model of univalent foundations and then touring
Shulman's vast generalization, which provides an interpretation of homotopy
type theory with strict univalent universes in any $\infty$-topos. As we will
explain, this achievement was the product of a community effort to abstract and
streamline the original arguments as well as develop new lines of reasoning.
