This paper provides a comprehensive review of Extended/Generalized Finite Element Methods (XFEM/GFEM) for material modeling, focusing on their applications to problems involving fractures, dislocations, grain boundaries, and phase interfaces. XFEM and GFEM are versatile techniques that simplify the modeling of discontinuities, singularities, and complex geometries in materials science. The methods allow for independent meshing, which is particularly useful for problems with evolving geometries, such as growing cracks or moving phase boundaries. The paper discusses the mathematical foundations of XFEM/GFEM, including the partition of unity concept, and provides detailed formulations for various applications. It also reviews the literature on these methods, highlighting recent advancements and challenges, such as quadrature issues and the treatment of cohesive cracks and dislocations. The paper emphasizes the advantages of XFEM/GFEM in handling complex geometries and discontinuities, making them valuable tools for advanced material modeling.This paper provides a comprehensive review of Extended/Generalized Finite Element Methods (XFEM/GFEM) for material modeling, focusing on their applications to problems involving fractures, dislocations, grain boundaries, and phase interfaces. XFEM and GFEM are versatile techniques that simplify the modeling of discontinuities, singularities, and complex geometries in materials science. The methods allow for independent meshing, which is particularly useful for problems with evolving geometries, such as growing cracks or moving phase boundaries. The paper discusses the mathematical foundations of XFEM/GFEM, including the partition of unity concept, and provides detailed formulations for various applications. It also reviews the literature on these methods, highlighting recent advancements and challenges, such as quadrature issues and the treatment of cohesive cracks and dislocations. The paper emphasizes the advantages of XFEM/GFEM in handling complex geometries and discontinuities, making them valuable tools for advanced material modeling.