Our original paper (Grosz, Joshi, and Weinstein, 1983) on centering claimed that certain entities mentioned in an utterance were more central than others and that this property imposed constraints on a speaker's use of different types of referring expression. Centering was proposed as a model that accounted for this phenomenon. We argued that the compatibility of centering properties of an utterance with choice of referring expression affected the coherence of discourse. Subsequently, we expanded the ideas presented therein. We defined various centering constructs and proposed two centering rules in terms of these constructs. A draft manuscript describing this elaborated centering framework and presenting some initial theoretical claims has been in wide circulation since 1986. This draft (Grosz, Joshi, and Weinstein 1986, hereafter, GJW86) has led to a number of papers by others on this topic and has been extensively cited, but has never been published. We have been urged to publish the more detailed description of the centering framework and theory proposed in GJW86 so that an official version would be archivally available. The task of completing and revising this draft became more daunting as time passed and more and more papers appeared on centering. Many of these papers proposed extensions to or revisions of the theory and attempted to answer questions posed in GJW86. It has become ever more clear that it would be useful to have a "definitive" statement of the original motivations for centering, the basic definitions underlying the centering framework, and the original theoretical claims. This paper attempts to meet that need. To accomplish this goal, we have chosen to remove descriptions of many open research questions posed in GJW86 as well as solutions that were only partially developed. We have also greatly shortened the discussion of criteria for and constraints on a possible semantic theory as a foundation for this work.