We present a general, consistency-based framework for belief change. Informally, in revising K by A, we begin with A and incorporate as much of K as consistently possible. Formally, a knowledge base K and sentence A are expressed, via renaming propositions in K, in separate languages. Using a maximization process, we assume the languages are the same insofar as consistently possible. Lastly, we express the resultant knowledge base in a single language. There may be more than one way in which A can be so extended by K: in choice revision, one such ``extension'' represents the revised state; alternately revision consists of the intersection of all such extensions. The most general formulation of our approach is flexible enough to express other approaches to revision and update, the merging of knowledge bases, and the incorporation of static and dynamic integrity constraints. Our framework differs from work based on ordinal conditional functions, notably with respect to iterated revision. We argue that the approach is well-suited for implementation: the choice revision operator gives better complexity results than general revision; the approach can be expressed in terms of a finite knowledge base; and the scope of a revision can be restricted to just those propositions mentioned in the sentence for revision A.