  We survey the developments in the model theory of tracial von Neumann
algebras that have taken place in the last fifteen years. We discuss the
appropriate first-order language for axiomatizing this class as well as the
subclass of II$_1$ factors. We discuss how model-theoretic ideas were used to
settle a variety of questions around isomorphism of ultrapowers of tracial von
Neumann algebras with respect to different ultrafilters before moving on to
more model-theoretic concerns, such as theories of II$_1$ factors and
existentially closed II$_1$ factors. We conclude with two recent applications
of model-theoretic ideas to questions around relative commutants.
