An efe cient formulation for the inverse and direct design optimization of airfoils in laminar and turbulent e ow is presented. Our procedure simultaneously relies on converging the design process and the analysis, while sequentiallyusingprogressivelye nergrids.Alle owanalysesandtheobjectivefunctionarecomputedwithstandard accurateviscouse owsolvers.Approximatedesignsensitivitiesarecomputedusingahighlyefe cientadjointsolution procedurebased onanartie cially dissipative, inviscidauxiliarye owsolverona coarsegrid.Ourprocedureinvolves what we term progressive optimization, whereby a sequence of operations, containing a partially converged e ow solution, followed by a partially converged adjoint solution followed by an optimization step, is performed. This approach has been tested on several sample inverse and direct (constrained ) design problems involving twodimensional airfoils in laminar and turbulent e ow conditions. The methodology has exhibited robustness and was shown to be highly efe cient, with converged design optimizations produced in no more than the computational work to perform two e ow analyses on the e nest mesh, independent of the number of design variables.