How many pairs of roots of unity (x,y) are there such that x+y=1? What about x^5=y^7? The nature of these kinds of questions is determined by the Manin--Mumford Conjecture, which concerns the solvability of systems of polynomial equations in roots of unity. There are by now several different proofs of the Manin--Mumford Conjecture; in this talk I will sketch one due to Jonathan Pila and Umberto Zannier, based on counting intersection points with the aid of a remarkable theorem from model theory.