The diffraction of high frequency torsion waves by disc-shaped obstacles, situated in solids which are homogeneous, isotropic and of infinite extent, are considered in this thesis. In a high frequency limit these problems are formulated as Fredholm integral equations of the second kind. The thesis is divided into two chapters:- Chapter I: diffraction of high frequency torsion waves by a penny-shaped crack. Explicit asymptotic expressions are obtained for the dynamic stress intensity factors and the scattering coefficients. These results predict an oscillatory behaviour of the stress intensity factors at high frequencies. Chapter II: diffraction of high frequency torsion waves by a rigid disc. Explicit asymptotic expressions are obtained for the torque resisting the motion of the disc, and for the scattering coefficients. In both chapters extensive, numerical results are given.