Mean flow and turbulence measurements have been made in three-dimensional turbulent boundary layers in curved ducts, simulating adverse pressure gradients on two ‘infinite’ swept curved wing surfaces with concave and convex curvature respectively. The ratio of the initial boundary-layer thickness to the surface radius of curvature in both cases is approximately 0.01, the value used in the earlier two-dimensional turbulent boundary-layer studies on the effects of concave and convex curvature by Hoffmann, Muck & Bradshaw (1985) and Muck, Hoffmann & Bradshaw (1985) respectively. The pressure-driven crossflow has nearly the same streamwise distribution as in the ‘infinite’ swept flat-surface experiment of Bradshaw & Pontikos (1985), which used a similar duct. The results of the present study show that the coupled effects of mean flow three-dimensionality and prolonged mild surface curvature of either sign have rather a weak influence on the turbulence structure, unlike the significant influence of the above extra strain rates when applied individually. In the concave case, the effect of the crossflow appears to oppose the destabilizing effect of curvature in addition to suppressing spanwise wavy inhomogeneities In contrast, the weak combined influence of convex curvature and crossflow, both of which, separately, tend to attenuate turbulence, implies that the interaction between the two effects is grossly nonlinear. Implications of the present results for turbulence modelling are briefly discussed.