The border between Lorraine and the neighbouring countries (Belgian Luxembourg, Grand Duchy of Luxemburg, Saar, Palatinat) knows deep changes since the practical application of the Single market in 1993 and the development of border work towards Luxemburg. The transformations appear initially on a morphological level: border-effects are diffused more and more towards the interior of the territory at the point to draw a «trans-border hem» which runs, along the main major routes, to the doors of the agglomeration of Metz. The border between States also evolves in its functions: with the multiplication of the exchanges, it seems a new marker of the social and tax differences within the European Union between close and foreign communities but also inside the same community. Trans-border mobility thus induce new forms of territorial discontinuities. Beyond the geographical approach, they reveal us the stakes of social Europe to build.