The idea for this special issue first arose at the Fourth Workshop on Uncertainty in Artificial Intelligence (AI) held at Minneapolis, Minnesota, in July 1988. Jim Bezdek, the editor-in-chief of this journal, asked us if we were willing to act as guest editors. We thought it was a great idea for two reasons: (1) There was (and still is) considerable interest in the AI community on the Dempster-Shafer theory of belief functions, and (2) members of the AI community seemed to like the basic premises of the belief function approach but were not quite sure how to use them in real applications. As first-time guest editors usually do, we made up an ambitious schedule, which has since slipped quite a bit. However, thanks to Jim Bezdek, we will still make the targeted 1990 publication date. A lot of our gratitude in making this issue a reality goes to the referees for their thorough reviews and prompt responses. We thank them by name at the end of this introduction. With their help, we accepted the papers by Tom Strat, Didier Dubois and Henri Prade, and Greg Provan--for inclusion here. In addition, we invited Glenn Sharer and Judea Pearl to write position papers that would be published unrefereed. Both kindly consented to do so and, as this issue testifies, delivered on their promises. These two papers make significant contributions to the state of the art in belief function theory, semantics, and application.