| Résume||The model theoretic analysis of the j-function has taken two apparently different paths in recent years. One of these has been inspired by arithmetic geometric considerations, has been done in infinitary logic and has produced some categoricity results (Harris). The other one (due initially to Freitag and Scanlon, later Aslanyan) has focused on the differential equations satisfied by the j-function. I will describe these two approaches, as well as some recent lines of generalization. The last part is joint work with Alex Cruz.