We discuss the Manin-Mumford conjecture (about the closure of any set of torsion points in an abelian variety), the André-Oort conjecture (about the closure of any set of CM-points in a moduli space) and the Hecke Orbit Conjecture (about the closure of the Hecke orbit of a moduli point). These conjectures, on the borderline of geometry and arithmetic, seem to be (have been) solved. We explain the similarities. We will discuss the motivation for these conjectures, and in some cases we will say something about methods of proofs.