The boundary integral equation is one of several equivalent forms of governing equations that can be used to compute approximate solutions to boundary value problems in elasticity and potential flow analysis. Since it determines the entire solution in terms of values only on the boundary, there are possible order-of-magnitude advantages in solution time and geometric complexity over better known domain-based' methods such as finite elements and finite differences. In practice, it has been hard to capitalize on these advantages. Many of the difficulties center around inablility to perform certain numerical integrations. This thesis presents (a) a systematic modal' method of converting singular integrals to easier integrals over far' surfaces (b) an otimal quadrature method for the nearly singular' integration problem. An existence proof is given to show that all surface integrals arising from the 3D boundary integral can be converted to easier contour integrals if basis functions are constructed in a cartesian sense, rather than the common parametric formulations. Strokes vectors needed to make this result useful are demonstrated for the Laplace equation and for some cases elasticity. Comparison to analytic benchmark cases shows that the method produces accurate stress intensity factors for 3-dimensional fracture analysis.