Abstract: A continuous theory T of bounded metric structures is said to be kappa-categorical if T has a unique model of density kappa. Work of Ben Yaacov and Shelah+Usvyatsov shows that Morley's Theorem holds in this context: if T has a countable signature and is kappa-categorical for some uncountable kappa, then T is kappa-categorical for all uncountable kappa. In classical (discrete) model theory, one has a strong structure theorem for uncountably categorical theories T, due to Baldwin+Lachlan: there is a strongly minimal set D defined over the prime model of T such that every uncountable model M of T is minimal and prime over D(M). In the more general metric structure setting, nothing remotely like this is known. Indeed, the metric analog of a strongly minimal set is nowhere to be seen, at the moment. If one restricts attention to metric structures based on (unit balls) of Banach space structures, more is known. The appropriate analog of strongly minimal sets seems to be the unit balls of Hilbert spaces. After the speaker called attention to this phenomenon in some examples from functional analysis, Shelah and Usvyatsov investigated it and proved a remarkable result: if M is a nonseparable Banach space structure (with countable signature) whose theory is uncountably categorical, then M is prime over a Morley sequence that is an orthonormal Hilbert basis of length equal to the density of M. There is a wide gap between this result and verified examples of uncountably categorical Banach spaces, which leads to the question: can a stronger such result be proved, in which the connection to Hilbert space structure is clearly expressed in the geometric language of functional analysis? Here the ultimate goal would be to prove analogues for Banach space structures (or even for general metric structures) of the Baldwin-Lachlan Theorems. The recent progress discussed in this talk concerns new examples of uncountably categorical Banach spaces (of which there have been very few previously known). This is joint work with Yves Raynaud (Univ. of Paris 6). We also have some ideas about where to look for uncountably categorical Banach space structures with richer signatures, and some conjectures about settings where such structures cannot exist.