The theory of distal measure-preserving actions of countable groups was initiated by Zimmer and was used by Furstenberg in his ergodic-theoretic proof of Szemerédi's theorem. I am going to discuss a recent joint work with Tomás Ibarlucía in which we prove some rigidity results for strongly ergodic, distal actions, generalizing earlier results of Ioana and Tucker-Drob. Perhaps the most interesting feature of our approach is the use of continuous logic: a model-theoretic framework adapted to the study of metric structures.