An anthology is a book of stories by different authors, as chosen and gathered by the book's editor(s). (This is in contrast to a collection, a book of stories by a single author, as chosen by the author or perhaps a separate editor. The distinction between the two terms is common, though not universal, in the science fiction, fantasy, and horror fields, while outside the genres the terms are often used interchangeably.)
Over the course of the past century, especially since the 1940s, tens of thousands of anthologies have been published in the science fiction, fantasy, and horror fields. Isfdb's stats page indicates some 625,000 publications in its database, of which about 29,000 are anthologies.
This section of sfadb.com profiles some 1420 sf/f/h anthologies, grouped onto 118 pages defined by editor or theme. Just as the awards section of this site makes no attempt to compile every single award in the news (as at Locus Online: Awards), the compilation here doesn't attempt to compile every anthology ever published.
The criteria for inclusion of anthologies here is to parallel the compilation of awards.
First, anthologies with the broadest scopes: Halls of Fame, and expert editors' selections of all-time best stories at their times (as in those books by Healy/McComas, Silverberg, Hartwell, Dozois, et al.);
Second, anthologies by the major anthologists in these fields over the decades: Conklin, Derleth, Wollheim, Merril, Pohl, Knight, Silverberg, Carr, and others.
Third, anthologies on principle themes in the history of sf/f/h, including the various "theme" anthologies, and those serving as textbooks, about cyberpunk, about war & peace, about sex & gender, and those that capture the best of the major magazines over the decades.
And Fourth, initially disregarded in the unposted version 1.5 of this compilation (since they don't contribute to the assessment and ranking, combined with awards data, of all-time short fiction, which is forthcoming), the major series of original anthologies, by Pohl, Knight, Silverberg, Carr, and others. They became included so that these pages would serve as a history of anthologies, and indirectly, of the sf/f/h fields they reflect.
Links on the upper right of these pages enable scrolling through them chronologically, in effect presenting a history of these anthologies, and the underlying themes of sf/f/h, from the 1940s to present. Each of the 118 group pages includes a composite table of contents of all the books in that group, in many cases supplemented by tallies of authors and/or sources. Total descriptive text on the 118 pages is about 30,000 words. And most pages show photos of many of the actual books (not just cover scans copied from other sites), as in the photo at the top of this page. (Note that on most of these pages, you can view images full-sized using whatever commands work with your browser.)
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Anthologies are significant for understanding the history of the fantastic genres, since science fiction, fantasy, and horror appeared in short stories before they became popular in novels. Moreover, most early genre short stories were published in cheap magazines (e.g. pulps), in the second quarter of the 20th century, and such magazines were ephermeral; anthologies, especially in hardcover beginning in the 1940s, became ways of preserving those stories in more permanent form.
Anthologies have thus served two functions: they have preserved stories for later generations that otherwise might have been lost in moldering magazines; and the selection of the stories for anthologies has served as validation, by repeated editorial judgement, of the stories' relative excellence compared to all the others that have been forgotten.
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This compilation serves two functions. The data from reprint anthologies supplements the awards data, identifying significant stories published before the advent of the genre awards, and recent worthwhile stories other than the relatively few that are nominated for awards. And the aggregate of these profiles, of both reprint and original anthologies, comprises a history not just of anthologies, but indirectly of the genres of science fiction, fantasy, and horror themselves.