Abstract: An operator space is an operator-norm closed linear subspace of B(H), the C* algebra of bounded linear on a Hilbert space H. For reasons that will be explained in the talk, an operator space can be viewed as a noncommutative generalization of a Banach space. A fundamental result of Junge and Pisier states that there are many more operator spaces than there are Banach spaces. More precisely, for any n at least 3, the set of n-dimensional operator spaces, when equipped with the strong topology, is not separable. As a corollary, there is no universal separable operator space. In this talk, I will mention how the techniques of Junge and Pisier can be used to prove a model-theoretic result, namely that the class of so-called 1-exact operator spaces is not an omitting types class. I will mention a descriptive-set theoretic strategy for removing this dependence on the techniques of Junge and Pisier, which would be desirable as I will then show how the fact that the! 1-exact operator spaces are not an omitting types class, together with a model-theoretic analysis of the theory of the Gurarij operator space, can be used to give a purely model-theoretic proof of the fact that there is no universal separable operator space. This is joint work with Thomas Sinclair and separately with Martino Lupini.