I will briefly introduce continuous model theory and von Neumann algebras and explain some of the connections between them. Continuous model theory for metric structures is a version of model theory suitable for analysis, where the structures are metric spaces and the predicates take real values rather than true and false. This continuous model theory has had several connections with the study of von Neumann algebras, certain rings of operators on Hilbert space that serve as a non-commutative analog of measure spaces. I will then speak about joint work with Farah and Pi discussing which von Neumann algebras have the model-theoretic properties of quantifier elimination and model completeness. We show many von Neumann algebras do not have these properties, which makes them difficult and interesting objects of study.