The study and prediction of crack behavior in materials have always been central to classical theories of solids. Since Griffith's 1921 seminal paper, all subsequent literature on brittle fracture has built on his work. In this thesis, we focus on the extended variational setting of Francfort and Marigo, which aims at studying quasi-static brittle fracture by incorporating an interplay between elastic and fracture energies and a minimization over admissible crack evolutions. The assumptions and limitations of the model are clearly stated, as are the similarities and differences with Griffith's model. Some existence results are needed for an appropriate formalization in the context of the calculus of variations.