This thesis is concerned with the dev~lopment and application of a general class of boundary collocation te9hniques in relation to the solving of boundary value problems of elasticity. A common feature of the techniques is that the body is embedded in a fictitious region and suitable singularities are introduced either on the boundary of the real body or in the fictitious region outside it. The particular solutions of the elasticity equations of the problem corresponding to these singularities are then superposed and their intensities are determined so that the boundary condition ~s satisfied in an approximate manner. One such technique, the point matching technique, is described in Chapter 1 and applied successfully in Chapter 2 to obtain the stress concentration around the intersection of two cylindrical holes in an . infinite elastic body. Since the success of this method depends to a large extent on the judgement used in choosing the basic solutions and the matched points, in the next two chapters an attempt is made to develop a simple but efficient alternative to the point matching technique. Methods with singularities distributed on the boundary are investigated in Chapter 3 and those with singularities situated in the region outside are considered in Chapter 4. Comparing tMse methods on the basis of three simple test problems the boundary segment method (method 2 of Chapter 4) is justified as the best. This is then compared, in Chapter 5, with the finiteaement method on the basis of two complex problems of stress analysiso Finally, it is concluded that the boundary segment method is capable of providing a reasonably rapid and accurate analysis of boundary value problems of elasticity. It is particularly suited to problems of stress concentration, has specific advantageS'over th~'finite element method When applied to _ such problems and the judgement required to solve any particular case is never ~re than that necessary in the finite element method.