Abstract: We introduce a framework in which classes of complete metric spaces with continuous extra-structure (e.g. Banach spaces) can be treated. It can be viewed in several equivalent ways, each one involves introducing a generalized logic, which satisfies the compactness theorem, but does not have the same expression power as the first order logic.

We develop the basics of model theory (stability theory) in this context and present the analog of Morley's categoricity theorem for classes of complete metric spaces. This work is a part of my PhD thesis which is being done under the supervision of S.Shelah.