The AGM theory is the dominating paradigm in the field of belief change but makes some non-elementary assumptions which disallow its application to certain logics. In this paper, we recast the theory by dropping most such assumptions; we determine necessary and sufficient conditions for a logic to support operators which are compatible with our generalized version of the theory and show that our approach is applicable to a broader class of logics than the one considered by AGM. Moreover, we present a new representation theorem for operators satisfying the AGM postulates and investigate why the AGM postulates are incompatible with the foundational model. Finally, we propose a weakening of the recovery postulate which has several intuitively appealing properties.