AbstractOn the basis of the Leonov viscoelastic constitutive equation, oscillatory shear flow of elastic fluids in the linear and nonlinear regimes has been considered. The Fourier components and associated phase angles of the shear and normal components of the elastic strain tensor have been found as functions of frequency and deformation amplitude in the range usually employed in experiment, and are presented in a form convenient for further rheological applications. In the linear case, the results correspond to many known theories. In the nonlinear case, the theoretical results have been compared with experiments, on different polymeric systems, with very good agreement being obtained for the shear stress in polymeric solutions but only qualitative agreement for the shear stress and first normal‐stress difference in polymer melt