Axiomatic classes of models in modal logics

In the model theory for any formal language, the results called definability criteria play a fundamental role. These are theorems that give necessary and sufficient conditions for an arbitrary class of structures to be (finitely) axiomatizable (in the chosen language) or to be representable as the union of (finitely) axiomatizable classes (these four options are exhaustive in some sense). A prominent result of this kind is Keisler's theorem saying that a class of first-order structures is elementary if and only if it is closed under elementary equivalence and ultraproducts. In this talk we give a survey of known and new results of this kind for some modal languages and classes of (pointed) Kripke models (not frames).