This thesis introduces an alternative method to evaluate Stress Intensity Factors (SIFs) in computational fracture mechanics directly, using the Extended Dual Boundary Element Method (XBEM) for 2D problems. A novel auxiliary equation introduced which enforces displacement continuity at the crack tip to yield a square system. Additionally, the enrichment method has been extended to 3D, so that the J-integral with XBEM and a direct technique are used to evaluate SIFs. This includes a complete description of the formulation of enrichment functions, a substitution of the enriched form of displacement into boundary integral equations, treatment of singular integrals, assembly of system matrices and the introduction of auxiliary equations to solve the system directly. The enrichment approach utilizes the Williams expansions to enrich crack surface elements for accurate evaluation of stress intensity factors. Similar to other enrichment methods, the new approach can yield accurate results on coarse discretisations, and the enrichment increases the 2D problem size by only two degrees of freedom per crack tip. In the case of 3D, the number of the new degrees of freedom depends on the desired number of crack front points where SIFs need to be evaluated. The auxiliary equations required to yield a square system are derived by enforcing continuity of displacement at the crack front. The enrichment approach provides the values of singular coefficients KI, KII and KIII directly in the solution vector; without any need for postprocessing such as the J-integral. Numerical examples are used to compare the accuracy of these directly computed SIFs to the J-integral processing of both conventional and XBEM approximations.