Adaptive methods have not yet been applied to computational earthquake seismology. The main goal of this talk is to motivate their introduction into this field. After illustrating outstanding open questions in the dynamics of earthquake rupture, I will describe the prevailing modeling paradigm, where faults are represented by reactivated macroscopic shear cracks in linear elastic media, controlled by friction. The numerical methods currently in use in our community will be summarized and their limitations pointed out. Then, I will review arguments for the failure of some of the current assumptions and resurrect a rather unexplored paradigm, where the non linear rheology of the rock surrounding the fault is explicitly modeled, instead of being lumped into a surface friction law. These models seem to reconcile some of the observational paradoxes. The multi-scale nature of the problem will be emphasized in view of the potential application of adaptive numerical methods. I will finally mention other challenging problems in computational seismology that could benefit from adaptivity.