A unitary view of the most common formulations that have been proposed and used to model the movement and shape of forest fire fronts is given in this paper. The whole set of formulations is designated as the local quasi-equilibrium approach or model after the main physical hypothesis in which it is based. The physical and analytical aspects of the local quasi-equilibrium model are developed. Three main formulations based, respectively, on the Fermat principle, Huygens principle and the normal fire rate of spread, are given to the model and mutually compared. Some key considerations and proposals regarding the experimental test of the model are also presented. Finally, the main open issues are outlined.