This paper presents the analysis and implementation of the variational formulation of quasi-static brittle fracture mechanics proposed by G.A.~Francfort and J.-J.~Marigo in 1998. We briefly present the model itself, and its variational approximation in the sense of --convergence. We propose a numerical algorithm based on Alternate Minimizations and prove its convergence under restrictive assumptions. We establish a new necessary condition for optimality for the entire time evolution from which we derive the Backtracking algorithm. We give some elements of analysis of the Backtracking algorithm on a simple problem. We present realistic numerical simulation of a traction experiment on a fiber-reinforced matrix, and of the propagation of cracks in a perforated sample under mode-I loading.