In this paper, we identify the limiting stress tensor of a thin elastic shell whose thickness approaches zero. In the linear case, we use a convergence result established for the displacement field in order to get the convergence of the contravariant components of the linearised stress tensor. In the nonlinear case, the identification of the first Piola-Kirchhoff stress tensor is made through a formal asymptotic analysis. In both cases, we show that these limiting stress fields depend on the geometry of the shell and on the boundary conditions.