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The safety of engineering structures depends heavily on the presence of cracks, which may propagate and lead eventually to structural failure. This dissertation aims to advance the computational modeling of fracture, within the context of linear elastic fracture mechanics (LEFM) and cohesive zone models (CZMs). The extended finite element method (XFEM) is employed as the discretization method and cracks in both homogeneous and bimaterial solids are considered in this work. First, a novel set of enrichment functions within the framework of XFEM is proposed for the LEFM analysis of interface cracks in bimaterials. The motivation for the new enrichment set stems from the revelation that the accuracy of the widely accepted 12-fold bimaterial enrichment functions significantly deteriorates with the increase in material mismatch. To this end, we propose an 8-fold material-dependent enrichment set, derived from the analytical asymptotic displacement field, that well captures the near-tip oscillating singular fields of interface cracks, including the transition to weak discontinuities of bimaterials. The new enrichment set is tested on various examples and found to outperform the 12-fold set in terms of accuracy, conditioning, and total number of degrees of freedom (DOFs). The formulation is then extended to include high-order enrichment functions and accurate stress and displacement fields are obtained. The complex stress intensity factors (SIFs) of interface cracks are evaluated by employing Irwin's crack closure integral. To this end, a closed-form SIF formulation in terms of the enriched DOFs is derived by matching the leading term in the XFEM with an analytical expression of Irwin's integral. Hence, the SIFs of interface cracks can be directly obtained upon the solution of the XFEM discrete system without cumbersome post-processing requirements. The proposed method is shown to work well on several benchmark examples involving straight and curved interface cracks, giving accurate SIF results. Another contribution of the work is the application of Irwin's integral to the estimation of SIFs for curved homogeneous cracks. At the core, the proposed approach employs high-order enrichment functions to accurately capture the near-tip fields and evaluates the original definition of Irwin's integral through closed-form formulations in terms of enriched DOFs. An improved quadrature scheme using high-order isoparametric mapping together with a generalized Duffy transformation is proposed to integrate singular fields in tip elements with curved cracks. The proposed extraction approach is shown to yield decomposed SIFs with excellent accuracy and avoid the need for auxiliary fields as in J-integral method. Second, with respect to cohesive fracture, a discrete damage zone model (DDZM) is proposed following a rigorous thermodynamic framework similar to that of continuum damage mechanics (CDM). For the modeling of mixed-mode delamination, a novel damage evolution law is proposed to account for the coupled interaction between opening and sliding modes of interface deformations. A comprehensive comparison made with several popular CZMs in the literature demonstrates the thermodynamic consistency of the DDZM. The proposed interface model is integrated with the XFEM and the effectiveness of this framework has been validated on various benchmark problems. Finally, a novel continuous/discontinuous method is proposed to simulate the entire failure process of quasi-brittle materials: from the nucleation of diffuse damage to the development of discrete cracks . An integral-type nonlocal continuum damage model is coupled in this framework with DDZM with a new numerical energetic coupling scheme. The transition from the continuous (CDM) to the discontinuous approach (DDZM) can be triggered at any damage level with a weak energetic equivalence preserved. A few benchmark problems involving straight and curved cracks are investigated to demonstrate the applicability and robustness of the coupled XFEM cohesive-damage approach.