Over the last decade, the adjoint method has been consolidated as one of the most versatile and successful tools for aerodynamic design. It has become a research area on its own, spawning a large variety of applications and a prolific literature. Yet, some relevant aspects of the method remain relatively less explored in the literature. Such is the case with the adjoint boundary problem. In particular for Euler flows, both fluid dynamic and adjoint equations entail complementary Riemann problems, and these yield boundary conditions that are fully consistent with well-posedness. In the literature, this approach has been pursued solely in terms of Riemann variables. This work formulates the adjoint boundary problem so as to correspond precisely to that imposed on the flow, as it is given in terms of primitive variables. Test results have shown to be in agreement with the traditional approach for external flow problems. Copyright © 2012 John Wiley & Sons, Ltd.