The theory of brittle fracture, as first elaborated by Griffith in the 1920's, is still widely used in engineering practice. Yet it cannot initiate fracture, is powerless when trying to predict crack path and does not know how to handle sudden crack jumps. Around 1998, J.-J. Marigo and I proposed a model based on energy minimization which attempts to do away with those obstacles, while departing as little as feasible from Grifffith's theory. I will first describe the proposed model and evoke its specific shortcomings. From a mathematical standpoint, the model resembles a kind of evolutionary, constrained and vector-valued image segmentation problem in the sense of Mumford & Shah. C. Larsen and I have shown existence of a solution to the evolution for the weak -a la De Giorgi - formulation of the problem. I will very briefly describe the result and the method that was used in the proof; I will also mention the non-trivial extension to the case of a non-convex bulk energy, obtained in collaboration with G. Dal Maso and R. Toader. The model is readily amenable to numerics through various regularization of the energy which Gamma-converge to the original energy. This is mainly the work of B. Bourdin. I will present the main numerical algorithm, discuss the fascinating and multiple numerical issues and show Bourdin's latest 2 and 3-d. computations. If time permits, I will show how the adoption of a cohesive surface energy and of a more lenient minimization criterion may cure the shortcomings of the variational Griffith model. Unfortunately, the mathematical results are yet very primitive whenever one departs from the 1d setting.