Many manifolds M come naturally with a distinguished class of submanifolds exhibiting a special behaviour. For example Abelian varieties come with sub-Abelian varieties, Shimura varieties with sub-Shimura varieties, locally homogeneous spaces with totally geodesic sub-spaces and strata of translation surfaces with affine invariant submanifolds. The extended Zilber-Pink paradigm suggests the behaviour of the intersections between a submanifold of M and the special submanifolds by a simple dimension count. I will explain the power of this unifying viewpoint in two concrete situations. Namely the classical Noether-Lefschetz locus (joint with Klingler and Ullmo) and the more delicate case of translation surfaces (joint with D. Urbanik).