Many important problems in arithmetic geometry can be interpreted as questions regarding unlikely intersections. The Zilber-Pink conjecture on mixed Shimura varieties is a vast example. In this talk, we will focus on a special case - the Andre-Oort conjecture - which is now known to hold for many Shimura varieties thanks to new techniques from model theory. We will explain these techniques, as well as others from number theory, algebraic geometry and ergodic theory. To introduce the topic, we will discuss two analogues of the Andre-Oort conjecture - Lang's conjecture on roots of unity, and the Manin-Mumford conjecture on abelian varieties.