Appears in collection : 1923-2023, Centenaire de René Thom

This talk is about two papers of Rene Thom. Firstly, the famous one in the fifties in “Commentarii Mathematici Helvetici “ celebrated for its methods of transversality and cobordism, but where the originating problem or question related to the title is much less well known . Secondly, the one in the sixties “BAMS “ where the mysterious term “manifolds with singularities “ was completely revealed in terms of manifold strata with locally constant transversal structure. This , with examples and ideas revealing the appropriate context is rougher than smooth and as least as good as topological.