  We introduce and study the framework of compact metric structures and their
associated notions of isomorphisms such as homeomorphic and bi-Lipschitz
isomorphism. This is subsequently applied to model various classification
problems in analysis such as isomorphism of C*-algebras and affine
homeomorphism of Choquet simplices, where among other things we provide a
simple proof of the completeness of the isomorphism relation of separable,
simple, nuclear C*-algebras recently established by M. Sabok.
