This work is devoted to the study of models of fractures growth in brittle elastic materials; it collects the results obtained during my Ph.D., that are contained in [77, 76, 78]. We consider quasi-static rate-independent models, as well as rate-dependent ones and the case in which the first ones are limits of the second ones when certain physical parameters vanish. The term quasistatic means that, at each instant, the system is assumed to be in equilibrium with respect to its time-dependent data; this setting is typical of systems whose internal time scale is much smaller than that of the loadings. By rate-independent system we mean that, if the time-dependent data are rescaled by a strictly monotone increasing function, then the system reacts by rescaling the solutions in the same manner.
