The model theory of j-functions (and more generally of modular and Shimura curves) has been studied by Adam Harris and Christopher Daw. They connected in an intriguing way the categoricity of (an infinitary theory of) the j-function with results in arithmetic geometry (a version of the Mumford-Tate conjecture). I will discuss some of these connections and the questions they raise for model theory, especially in connection with the quest for new versions of j-functions (e.g. on real fields).