There are deep connections between model theory of the infinitary logic Lω1ω and descriptive set theory: Scott analysis, the López-Escobar theorem or the Suzuki theorem are well known examples of this phenomenon. In this talk, I would like to present results of an ongoing research devoted to generalizing these connections to the setting of continuous infinitary logic and Polish metric structures. In particular, I will discuss a continuous counterpart of a theorem of Hjorth and Kechris characterizing essential countability of the isomorphism relation on a given Borel class of countable structures. As an application, I will give a short model-theoretic proof of a result of Kechris saying that orbit equivalence relations induced by continuous actions of locally compact Polish groups are essentially countable. This is joint work with Andreas Hallbäck and Todor Tsankov.