The functional analytic setting of various variational models in Fracture Mechanics requires the use of classes of functions with set of discontinuities of codimension one. The difficulty of finding good discretization for such classes of functions makes the direct numerical simulation of those variational problems challenging and highly problematic. For this reason, numerous regularizations have been proposed, the most successful of which are phase-field functionals. These elliptic regularizations were first introduced and analyzed in the work of Ambrosio and Tortorelli for the Mumford-Shah energy in image segmentation, inspired by a now classical example in phase transition by Modica and Mortola. Ambrosio and Tortorelli type approximations have become very popular both in the communities of Calculus of Variations and of Computational Mechanics to address a number of problems in applied sciences, especially in brittle fracture. In the talk, we will comment on some of those phase-field models, starting with Ambrosio and Tortorelli's, which eventually led to a useful variant for approximating cohesive energies in Fracture Mechanics.