Streamline curvature in the plane of the mean shear produces large changes in the turbulence structure of shear layers, usually an order of magnitude more important than normal pressure gradients and other terms in the mean-motion equations for curved flows. The effects on momentum and heat transfer in boundary layers are noticeable on typical wing sections and are very important on highly-cambered turbomachine blades: turbulence may be nearly eliminated on highly-convex surfaces, while on highly-concave surfaces momentum transfer by quasi-steady longitudinal vortices dominates the ordinary turbulence processes. The greatly enhanced mixing rates of swirling jets and the characteristic non-turbulent cores of trailing vortices are also consequences of the effects of streamline curvature on the turbulence structure. A progress report, comprises a review of current knowledge, a discussion of methods of predicting curvature effects, and a presentation of principles for the guidance of future workers.