This paper provides a review of the Extended Finite Element Method (XFEM) and Generalized Finite Element Method (GFEM) with an emphasis on their applications in material modeling. These methods are particularly useful for problems involving discontinuities, such as fracture, dislocations, grain boundaries, and phase interfaces. The key advantage of XFEM and GFEM is that they allow for the modeling of complex geometries without requiring the mesh to conform to the features of interest, such as cracks or grain boundaries. This is especially beneficial when combined with level set methods, which enable the simulation of evolving geometries like growing cracks or moving phase interfaces. The methods are based on the partition of unity concept, which allows for the enrichment of the finite element approximation space with functions that capture the behavior near discontinuities. This enrichment is done by introducing additional basis functions that account for the specific features of the problem, such as the asymptotic solutions near crack tips or dislocation cores. This approach reduces the need for mesh refinement in these regions, leading to more efficient computations. XFEM and GFEM have been applied to a wide range of problems, including the modeling of cracks, dislocations, grain boundaries, and phase interfaces. The methods have been shown to be effective in capturing the behavior of these features, even in complex geometries. The discrete equations for these methods are derived from the principle of virtual work and involve stiffness matrices that account for both the standard finite element terms and the enriched terms. The solution process involves solving these equations to determine the displacements and other variables of interest. The paper also discusses the challenges associated with the implementation of XFEM and GFEM, including the accurate evaluation of integrals in the weak form, which requires special quadrature techniques for singular and discontinuous functions. Additionally, the paper reviews the literature on the application of these methods to various problems, highlighting their effectiveness in modeling discontinuous phenomena in materials science. The methods have been successfully applied to problems such as cohesive crack modeling, dislocation dynamics, and grain boundary modeling, demonstrating their versatility and robustness in handling complex material behaviors.This paper provides a review of the Extended Finite Element Method (XFEM) and Generalized Finite Element Method (GFEM) with an emphasis on their applications in material modeling. These methods are particularly useful for problems involving discontinuities, such as fracture, dislocations, grain boundaries, and phase interfaces. The key advantage of XFEM and GFEM is that they allow for the modeling of complex geometries without requiring the mesh to conform to the features of interest, such as cracks or grain boundaries. This is especially beneficial when combined with level set methods, which enable the simulation of evolving geometries like growing cracks or moving phase interfaces. The methods are based on the partition of unity concept, which allows for the enrichment of the finite element approximation space with functions that capture the behavior near discontinuities. This enrichment is done by introducing additional basis functions that account for the specific features of the problem, such as the asymptotic solutions near crack tips or dislocation cores. This approach reduces the need for mesh refinement in these regions, leading to more efficient computations. XFEM and GFEM have been applied to a wide range of problems, including the modeling of cracks, dislocations, grain boundaries, and phase interfaces. The methods have been shown to be effective in capturing the behavior of these features, even in complex geometries. The discrete equations for these methods are derived from the principle of virtual work and involve stiffness matrices that account for both the standard finite element terms and the enriched terms. The solution process involves solving these equations to determine the displacements and other variables of interest. The paper also discusses the challenges associated with the implementation of XFEM and GFEM, including the accurate evaluation of integrals in the weak form, which requires special quadrature techniques for singular and discontinuous functions. Additionally, the paper reviews the literature on the application of these methods to various problems, highlighting their effectiveness in modeling discontinuous phenomena in materials science. The methods have been successfully applied to problems such as cohesive crack modeling, dislocation dynamics, and grain boundary modeling, demonstrating their versatility and robustness in handling complex material behaviors.