I will describe recent work with Ngaiming Mok and Jacob Tsimerman giving analogues of ``Ax-Schanuel'' for Shimura varieties. I will sketch  the role of ``o-minimal structures''in the proof, in particular point counting and powerful algebraicity results of Peterzil-Starchenko, in combination with hyperbolic geometry and monodromy, and describe versions in the setting of a differential field.