We will discuss two applications of infinitary continuous logic to complexity of equivalence relations. We will characterize in model-theoretic terms essentially countable isomorphism relations on Borel classes of locally compact Polish metric structures. This gives a new proof of Kechris' theorem that orbit equivalence relations of actions of Polish locally compact groups are essentially countable. We will also show that isomorphism on such classes is always Borel reducible to graph isomorphism. This immediately answers a question of Gao and Kechris whether isometry of locally compact Polish metric spaces is reducible to graph isomorphism. The first result is joint work with Andreas Hallbäck and Todor Tsankov.