Hodge theory and atypical intersections

Given a variation of Hodge structures $V$ over a smooth quasi-projective base $S$, I will explain the notion of an atypical subvariety for $(S, V)$ and state a simple general conjecture about these atypical subvarieties, as well as some results toward it. When $S$ is a Shimura variety and $V$ a standard variation of Hodge structure on $S$, one recovers the Zilber-Pink conjectures, in particular the Andr\'e-Oort conjecture.