This is the introductory lecture to a mini-course (joint with Eugenia Rosu) about complex multiplication, at the CIRM Spring School 2023. It gives an overview of the basic theory of elliptic functions.

I will first recall the general expectations of Shimura, Langlands, and Kottwtiz on the shape of the zeta function of a Shimura variety, or more generally its etale cohomology. I will then report on some recent progress which partially fulfills these expectations, for Shimura varieties of unitary groups and special orthogonal groups. Finally, I will give a preview of some foreseeable developments in the near future.