  We develop an analogue of the classical Scott analysis for metric structures
and infinitary continuous logic. Among our results are the existence of Scott
sentences for metric structures and a version of the Lopez-Escobar theorem. We
also derive some descriptive set theoretic consequences: most notably, that
isomorphism on a class of separable structures is a Borel equivalence relation
iff their Scott rank is uniformly bounded below $\omega_1$. Finally, we apply
our methods to study the Gromov-Hausdorff distance between metric spaces and
the Kadets distance between Banach spaces, showing that the set of spaces with
distance $0$ to a fixed space is a Borel set.
