Antiplane shear is a type of deformation in which the only nonvanishing displacement component is the out-of-plane displacement, orthogonal to the (x,y) plane. The corresponding nonvanishing shear stresses are within that plane. For this type of deformation, the displacement is a harmonic function of (x,y), satisfying the Laplace's equation. We solve and discuss the problems of antiplane shear of a circular annulus, a concentrated line force along the surface of a half-space, antiplane shear of a medium weakened by a circular or an elliptical hole, and the problem of a medium strengthened by a circular inhomogeneity. The stress field near a crack tip under remote antiplane shear loading is derived, as well as the stress field around a screw dislocation in infinite and semi-infinite media. The stresses produced by a screw dislocation near a circular hole or a circular inhomogeneity in an infinite homogeneous medium, and the stresses produced by a screw dislocation in the vicinity of a bimaterial interface are examined.