The paper reviews the Extended Finite Element Method (XFEM) and Generalized Finite Element Method (GFEM) with a focus on their applications in material science, particularly in modeling fracture, dislocations, grain boundaries, and phase interfaces. These methods are highlighted for their ability to handle complex geometries and discontinuities without requiring a mesh that aligns with these features. The authors discuss the partition of unity concept, which underpins both methods, and how it allows for the enrichment of the approximation space to capture singularities and discontinuities. The paper also covers the application of these methods to various problems, including: 1. **Cracks**: The methods are used to model crack propagation, with a focus on the use of Heaviside step functions and enrichment functions to capture the near-tip behavior. 2. **Dislocations**: Dislocations are modeled using a Volterra concept, where the body is cut across a surface and the relative displacements are prescribed according to the Burgers vector. 3. **Grain Boundaries**: The methods can model the complex geometries of grain boundaries without the mesh needing to align with these boundaries. 4. **Phase Interfaces**: XFEM and GFEM are combined with level set methods to track evolving interfaces, such as inclusions and phase transitions. The paper also addresses implementation issues, such as quadrature in XFEM/GFEM, and reviews relevant literature on these topics. The authors provide a comprehensive overview of the state-of-the-art in these methods and their applications, emphasizing their versatility and efficiency in solving complex material science problems.The paper reviews the Extended Finite Element Method (XFEM) and Generalized Finite Element Method (GFEM) with a focus on their applications in material science, particularly in modeling fracture, dislocations, grain boundaries, and phase interfaces. These methods are highlighted for their ability to handle complex geometries and discontinuities without requiring a mesh that aligns with these features. The authors discuss the partition of unity concept, which underpins both methods, and how it allows for the enrichment of the approximation space to capture singularities and discontinuities. The paper also covers the application of these methods to various problems, including: 1. **Cracks**: The methods are used to model crack propagation, with a focus on the use of Heaviside step functions and enrichment functions to capture the near-tip behavior. 2. **Dislocations**: Dislocations are modeled using a Volterra concept, where the body is cut across a surface and the relative displacements are prescribed according to the Burgers vector. 3. **Grain Boundaries**: The methods can model the complex geometries of grain boundaries without the mesh needing to align with these boundaries. 4. **Phase Interfaces**: XFEM and GFEM are combined with level set methods to track evolving interfaces, such as inclusions and phase transitions. The paper also addresses implementation issues, such as quadrature in XFEM/GFEM, and reviews relevant literature on these topics. The authors provide a comprehensive overview of the state-of-the-art in these methods and their applications, emphasizing their versatility and efficiency in solving complex material science problems.