Equations describing the flow within a boundary layer for elastic fluids are solved for several cases of physical significance. It is found that the question of increased or decreased wall stresses due to the presence of elasticity depends not only on the fluid, but on the particular geometry in question. THE SOLUTION of problems of engineering interest in the flow of elastic fluids requires that methods be developed for studying the properties of such materials in configurations other than the classical viscometric flow fields. It is unlikely that exact solutions of the equations of motion will be available, in general, for non-viscometric flows, and approximate methods will be necessary. Thus, several authors have considered the slow flow of an elastic fluid about a sphere by perturbation methods [1,2], and others have used integral approximations and perturbation methods to study the boundary layer flow about submerged objects [3, 41 and in the inlet to a tube [5]. In this paper the consideration of boundary layer flows is extended to include the entire class of Falkner-Skan ("wedge") flows for elastic fluids with shear dependent viscous and normal stress relations. A perturbation expansion in the Weissenberg number is always possible for weakly elastic fluids, and for several flows of interest similarity solutions may be obtained.