Eliashberg and Thurston discovered in '98 a surprising relationship
between foliations and contact structures in dimension 3: (most) C^2 foliations can be C^0-approximated by contact structures; there are moveover interesting connections between their properties. In higher dimensions, the best candidates for such type of results seem to be (codimension 1) foliations equipped with a conformal symplectic leafwise structure, according to recent results by Bertelson and Meigniez on manifolds with non-empty boundary. In this talk, I
will give an introduction to this and related problematics, and present a work joint with Lauran Toussaint where we prove the existence of conformal symplectic foliations on closed manifolds in any given almost contact class using the h-principle of Bertelson--Meigniez.