The talk will be devoted to the isomorphism relations on Borel classes of locally compact Polish metric structures. Using continuous logic, one can prove that they are always Borel reducible to graph isomorphism, which implies, in particular, that isometry of locally compact Polish metric spaces is reducible to graph isomorphism. This answers a question of Gao and Kechris. As a matter of fact, locally compact Polish metric structures behave very much like countable ones. For example, Hjorth, Kechris and Louveau proved that isomorphism of countable structures that is potentially of rank α+1α+1 multiplicative Borel class is reducible to equality on hereditarily countable sets of rank αα, and the same turns out to be true about locally compact Polish metric structures. Time permitting, I will also discuss certain variants of the Hjorth-isomorphism game, recently introduced by Lupini and Panagiotopoulos.