This work investigates and develops the extended finite element method (XFEM) for fracture mechanics. In fracture problems, a jump in the displacement field appears across the crack surface. Moreover, at the crack front a singularity can appear in the stress and strain fields. The advantage of the XFEM is in the mesh-independent approximation based on the enrichment of the approximation space. In order to capture high gradients and/or singularities that appear near the crack front, model-dependent enrichment functions are commonly used. Such functions are based on the asymptotic fields in the near-tip region of the fracture model. Consequently, for each fracture model, a different set of enrichment functions is required. The aim of this work is to find an approach that makes the XFEM really model-independent and, thereby, renders the development of crack-tip enrichment functions unnecessary.In this dissertation, a model-independent approach within the frame of the XFEM is realized based on the adaptive mesh refinement. The local mesh refinement is applied to ensure: (i) the ability to capture high gradients and/or singularities at the crack front and (ii) a high resolution at the crack surface. Herein, the adaptive mesh refinement leads to hanging nodes on the element edges and faces, in particular the if mesh is 1-irregular. Special conforming shape functions are used to ensure the conformity and the partition of unity property on these meshes, which is crucial for the application of XFEM. Proper integration rules are required to capture the interface within the elements with or without hanging nodes.The accuracy of the simulations is demonstrated by comparing results with analytical and numerical reference solutions. The proposed approach is implemented for static problems as well as for problems with propagating cracks within linear elastic and elasto-plastic fracture mechanics in two and three dimensions. Thereby, the effectiveness of the proposed approach to capture arbitrary high gradients is proven. The approach shows a large potential in problems where the exact analytical behavior at the crack front is unknown and, thus, enrichment functions may not be found successfully.