in a pioneering paper, pila and zannier showed how one can prove arithmetic results (the manin-mumford conjecture) using transcendental methods (the ax-lindemann conjecture). their approach has since been greatly developed, and is a major ingredient in the andre-oort conjecture for shimura varieties as well as the more general zilber-pink conjecture, that serves as a sort of flagship for the field of unlikely intersections. we'll explain this story and present a new result (joint with pila and mok) proving a general transcendence theorem known as ax-schanuel for arbitrary shimura varieties.