abstract: One implication of the Frobenius theorem states that there is no \(k\)-dimensional surface tangent to a non-involutive distribution of \(k\)-planes in \({\mathbb R}^n\). In this joint works with G. Alberti, A. Merlo and E. Stepanov, we explore some extensions of this result to contact sets and to currents. In particular, we wish to enlighten the trade between the regularity of the surface and the behaviour of the boundary of the contact set.