  An \'etale structure over a topological space $X$ is a continuous family of
structures (in some first-order language) indexed over $X$. We give an
exposition of this fundamental concept from sheaf theory and its relevance to
countable model theory and invariant descriptive set theory. We show that many
classical aspects of spaces of countable models can be naturally framed and
generalized in the context of \'etale structures, including the Lopez-Escobar
theorem on invariant Borel sets, an omitting types theorem, and various
characterizations of Scott rank. We also present and prove the countable
version of the Joyal-Tierney representation theorem, which states that the
isomorphism groupoid of an \'etale structure determines its theory up to
bi-interpretability; and we explain how special cases of this theorem recover
several recent results in the literature on groupoids of models and functors
between them.
