The principal objective has been to develope realistic elastic models of slip bands in polycrystalline materials to enable a more complete study to be made of the suggested mechanisms of yield propagation across grain boundaries. Existing models are reviewed in Chapter 1, special emphasis being given to those in which the slip band is represented by a continuous distribution of infinitesimal dislocations, an approximation used throughout the present work. A detailed description of the Wiener-Hopf technique for solving singular integral equations is given in Chapter 2, and is used in Chapter 3 to obtain the equilibrium distribution function of an array of screw dislocations inclined at certain angles to the plane interface between two different isotropic elastic half-spaces, and piled-up against this boundary under the influence of a uniform applied stress. Special attention is given in Chapter 4 to the case when the undislocated half-space is rigid, and to an infinitely thin notch in antipiane strain inclined at an arbitrary angle to a free surface. Solutions to certain boundary-value problems associated with elastically anisotropic half-spaces deformed in generalised plane strain are derived in Chapter 5, and used to investigate the interaction of dislocations with free surfaces and welded and freely-slipping interfaces between two such half-spaces. With the use of these results the analysis of Chapter 3 is generalised in Chapter 6 for cases when the half-spaces are elastically anisotropic, provided that they possess certain symmetry elements. The significance of the results is discussed in Chapter 7, where their application to the Hall-Petch relation for yield is considered in detail.