For many reasons, the Navier-Stokes equations have attracted the attention of a lot of researchers since the 30's. In spite of this, they still lead to several major open problems. In the first part of this talk, I will recall the formulations of some of them: the uniqueness, regularity and controllability problems in the 3D case. I will also mention some recent partial achievements. In the second part, I will present some contributions to the analysis, control and numerical solution of the equations.
 

 
