The present thesis proposes an innovative technique of applying enrichment to the Boundary Element Method to allow accurate analysis of 2D crack problems. An overview of fracture mechanics is given, with particular emphasis given to numerical methods and the techniques used to extract the highly important stress intensity factors - a measure of the singularity of a crack tip. The Boundary Element Method framework is described and later, the implementation of the new technique of enrichment is defined in detail. Finally, several crack problems are used to verify the accuracy of the method where the results are shown to compare very favourably with other well-established numerical methods.