We reconsider the problem of the boundary–layer flow of a non-Newtonian fluid whose constitutive law is given by the classical Ostwald–de Waele power–law model. The boundary–layer equations are solved in similarity form. The resulting similarity solutions for shear–thickening fluids are shown to have a finite–width crisis resulting in the prediction of a finite–width boundary layer. A secondary viscous adjustment layer is required in order to smooth out the solution and to ensure correct matching with the far–field boundary conditions. In the case of shear–thinning fluids, the similarity forms have solutions whose decay into the far field is strongly algebraic. Smooth matching between these inner algebraically decaying solutions and an outer uniform flow is achieved via the introduction of a viscous diffusion layer.