The flow of an incompressible second-order fluid due to torsional oscillations of an infinite plate when the fluid is infinite in extent as well as the case when it is bounded by another stationary parallel plate has been considered by expanding the velocity components and the pressure in powers of the amplitude of oscillation of the plate. In both cases the first-order solution consists of a transverse velocity and the second-order solution gives a radial-axial flow composed of a steady part and a fluctuating part. In the case of the unbounded plate the steady part of the radial flow does not vanish outside the boundary-layer region. Hence the equations are solved by another approximate method for the steady part of the flow. The effects of the non-Newtonian terms are to increase the non-dimensional boundary thickness and the shearing stress on the plate. In the case of two plates the velocity components and the shearing stresses on the plates have been expressed in powers of Reynolds number R for its small values. Their asymtotic behaviour for large R has also been studied. The asymtotic expansion of the fluctuating part of the radial-axial flow shows that the boundary layer is developed at both the plates.