The variational approach to fracture is presented in this work, offering a comprehensive overview of the mathematical and physical foundations of brittle fracture. The authors, Blaise Bourdin, Gilles A. Francfort, and Jean-Jacques Marigo, explore the concept of fracture as a competition between bulk and surface energy, a notion introduced by Griffith. They revisit Griffith's insight through the lens of variational calculus and incorporate Barenblatt's contributions, emphasizing the comparison of different surface energy models. The variational approach provides a framework for understanding crack initiation and propagation, highlighting the role of energy minimization and the dissipative nature of cohesive forces at crack surfaces. The text is structured into nine chapters, beginning with an introduction and a table of contents. The model is introduced in the first three sections, followed by an analysis of its impact on fracture mechanics in the next four sections. Numerical implementation is discussed in Section 8, while the extension of the model to fatigue is covered in Section 9. The authors emphasize the importance of mathematical modeling in fracture mechanics, acknowledging the need for a clear understanding of the physical principles underlying the model. They also highlight the challenges posed by non-convexity in the energetic landscape of fracture and the importance of the variational notion of metastability in resolving these issues. The authors acknowledge the support received from various institutions and individuals, and they express their gratitude for the contributions of others to the development of the variational approach to fracture. The work is presented as a contribution to the broader field of mechanics, with a focus on rate-independent processes. The authors hope that this work will provide a thorough and incisive understanding of the variational approach to fracture, offering a foundation for further research and development in the field.The variational approach to fracture is presented in this work, offering a comprehensive overview of the mathematical and physical foundations of brittle fracture. The authors, Blaise Bourdin, Gilles A. Francfort, and Jean-Jacques Marigo, explore the concept of fracture as a competition between bulk and surface energy, a notion introduced by Griffith. They revisit Griffith's insight through the lens of variational calculus and incorporate Barenblatt's contributions, emphasizing the comparison of different surface energy models. The variational approach provides a framework for understanding crack initiation and propagation, highlighting the role of energy minimization and the dissipative nature of cohesive forces at crack surfaces. The text is structured into nine chapters, beginning with an introduction and a table of contents. The model is introduced in the first three sections, followed by an analysis of its impact on fracture mechanics in the next four sections. Numerical implementation is discussed in Section 8, while the extension of the model to fatigue is covered in Section 9. The authors emphasize the importance of mathematical modeling in fracture mechanics, acknowledging the need for a clear understanding of the physical principles underlying the model. They also highlight the challenges posed by non-convexity in the energetic landscape of fracture and the importance of the variational notion of metastability in resolving these issues. The authors acknowledge the support received from various institutions and individuals, and they express their gratitude for the contributions of others to the development of the variational approach to fracture. The work is presented as a contribution to the broader field of mechanics, with a focus on rate-independent processes. The authors hope that this work will provide a thorough and incisive understanding of the variational approach to fracture, offering a foundation for further research and development in the field.