The Prandtl equation describes the motion of an incompressible fluid with low viscosity around an obstacle. When the flow is stationary, separation phenomena have been observed experimentally: there exists a point on the wall of the obstacle beyond which a reverse flow takes place close to the boundary. The purpose of this talk is to present some recent results related to this phenomenon. In collaboration with Nader Masmoudi, we proved that singularities develop close to the separation point. With Frédéric Marbach and Jean Rax, we studied the recirculation zone and proved the well-posedness of a toy-model describing this zone.