We introduce a new definition of stability, ε-stability, that implies local minimality and is robust enough for passing from discrete-time to continuous-time quasi-static evolutions, even with very irregular energies. We use this to give the first existence result for quasi-static crack evolutions that both predicts crack paths and produces states that are local minimizers at every time, but not necessarily global minimizers. The key ingredient in our model is the physically reasonable property, absent in global minimization models, that whenever there is a jump in time from one state to another, there must be a continuous path from the earlier state to the later along which the energy is almost decreasing. It follows that these evolutions are much closer to satisfying Griffith's criterion for crack growth than are solutions based on global minimization, and initiation is more physical than in global minimization models. © 2009 Wiley Periodicals, Inc.