In the mid 90's Eliashberg and Thurston established a fundamental link
between the more classical theory of (smooth!) foliations and that of contact topology
in dimension 3, which, amongst other things, played an important role in Mrowka and
Kronheimer’s proof of Property P Conjecture. Their theory gains its potency from the
fact that Gabai gave a very general method for constructing (smooth) taut foliations on
3-manifolds given from non-trivial homology classes.
On the other hand most foliations that occur in nature via (pseudo)-Anosov flows, surgery, gluing, blows ups... are not smooth in general. This naturally motivates the need to apply Eliashberg and Thurston’s theory to foliations of lower regularity. In this talk I will report on how their theory generalises. Time permitting I will discuss some applications and related questions.