Optimal aerodynamic shape design aims at finding the minimum of a functional by controlling the PDE modeling the flow using surface deformation techniques. As a solution to the enormous computational resources required for classical shape optimization of functionals of aerodynamic interest, probably the best strategy is to use in a systematic way, methods inspired in control theory. One of the key ingredients relies on the use of the adjoint system techniques to simplify the computation of

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... ents. In the present paper we will restrict our attention to optimal shape design in systems governed by the Navier-Stokes equations. We first review some standard facts on control theory applied to optimal shape design, and recall the Navier-Stokes equations in aerodynamical problems. We then study the adjoint formulation, providing a detailed exposition of how the derivatives of functionals may be obtained. Some relevant numerical issues will also be discussed. Finally, recent computational results applying a Level Set methodology are reviewed.