This paper presents a unified analysis of isotropic out-of-plane shear problems containing an arbitrary-shaped hole or an arbitrary rigid inclusion, and shows the closed-form analytical solutions. The applied forces considered in this paper are longitudinal shear stresses at infinity, point force, screw dislocation, dipole force, and dipole dislocation at an arbitrary point. The analysis is based on the complex-variable method using the conformal mapping technique. Using the results, the stress intensity factors (or singularity coefficients) for the tip of the cusp on the boundary L are given. Numerical examples of stress and displacement distributions are shown in graphical representations. The previous results published by several authors can be included as special cases of our solutions.