We survey some results in the model theory of metric structures related to different generalisations of the classic Ehrenfeucht–Fraïssé game. Namely, we look at a game of length $\omega$ that is used to characterise separable structures up to different notions of approximate isomorphism (such as linear isomorphisms between Banach spaces) in a framework that resembles that of positive bounded formulas. Additionally, we look at the (finite-length) EF game for continuous first-order logic and its variant of Ehrenfeucht's theorem. Last, we mention recent work on game comonads for continuous logic.
