A new explanation of the well documented effects of mild streamline curvature on the anisotropy of sheared turbulence has been developed. Its main underlying assumption is that the mean streamline curvature has no direct effect on the production and dynamics of each individual turbulent eddy, which is produced with the structure of purely sheared turbulence, and is subsequently convected downstream while retaining its initial anisotropy relative to fixed inertial coordinates. The local Reynolds stress anisotropy accumulates the contributions of all surviving eddies produced upstream, which, because the mean shear keeps changing direction, have different anisotropies, when viewed in terms of the local curvilinear coordinates; thus, the local anisotropy is influenced by the history of flow curvature only indirectly. A model developed to demonstrate the validity of the hypothesis requires only the specification of the turbulence anisotropies in a fully developed, rectilinear, reference flow (e.g., a rectilinear uniformly sheared flow), the geometrical features of the flow under study, and a dimensionless mean eddy lifetime. It predicts accurately the observed asymptotic turbulence structure of uniformly sheared flow subjected to prolonged, constant curvature and the exponential adjustment of this structure to stepwise changes in flow curvature. Predictions of the shear stress anisotropy in a curved mixing layer are also in good agreement with published data. In all these cases, the present model makes better predictions than two popular Reynolds stress models and a rapid distortion model. ©1998 American Institute of Physics.