An important question in ergodic theory dating back to the foundational paper of von Neumann is the so-called smooth realization problem: Are there smooth versions to the objects and concepts of abstract ergodic theory? Does every ergodic measure-preserving transformation have a smooth model?

One of the most powerful tools of constructing smooth volume-preserving diffeomorphisms with prescribed ergodic or topological properties is the Approximation by Conjugation method introduced by Dmitri Anosov and Anatole Katok. However, there are great challenging differences in the real-analytic case. In this talk, we give an overview of the method and its contributions to the smooth realization problem. We also present recent attempts to extend the AbC-method to the real-analytic category.