Propagation of two-dimensional transient out-of-plane shear waves in multilayered viscoelastic media is investigated. The multilayered medium consists of N different isotropic, homogeneous and linearly viscoelastic layers with more than one discrete relaxation time. The top surface of the layered medium is subjected to dynamic out-of-plane shear tractions; whereas, the lower surface is free or fixed. A numerical technique is employed to obtain the solution, which combines the Fourier transform with the method of characteristics. The numerical results are displayed in curves denoting the variations of the shear stresses with time at different locations. These curves reveal clearly the scattering effects caused by the reflections and refractions of inclined waves at the boundaries and at the interfaces of the layers. The curves also display the effects of viscous damping in the wave profiles. By suitably adjusting the material constants, the curves for the case of elastic layers are also obtained as a special case. The curves farther show that the numerical technique applied in this study is capable of predicting the sharp variations at the wave fronts. (C) 2001 Elsevier Science Ltd. All rights reserved.