This talk will cover the theoretical development of the Extended Symmetric Galerkin Boundary Element Method (X-SGBEM), a numerical framework proposed for 2D and 3D general crack growth simulations that allows the intersection of cracks with outer boundaries. This framework merges numerical concepts from Extended Finite Element Method (X-FEM)/Partition of Unity Method (PUM) and the SGBEM for linear elastic fracture simulations. Boundary Element Methods are preferred for linear elastic fracture calculations, as they are highly accurate and can be implemented into algorithms that are significantly faster than the frameworks developed for fracture mechanics using the Finite Element Method (X-FEM, phase-field/variational model for fracture). Boundary Element Methods have the advantage that only the boundaries and the cracks need to be modeled in the domain of interest; this facilitates minimal requirements for remeshing. Higher order crack tip elements can also provide precise fracture information (stress intensity factors) without requiring extremely fine meshing. Up until this day, there is no mathematically rigorous, optimized scheme that allows Boundary Element Methods to handle crack intersections with other surfaces (whether we are referring to other cracks or outer boundaries) without significant remeshing. Such a numerical technique could facilitate the study of fatigue crack growth and microcracking in brittle linear elastic materials, as well as simulations of fracture networks in geomaterials. This talk will include a review of prior developments of SGBEM in the context of fracture mechanics, and cover current developments of X-SGBEM.