The genesis of this anthology is quite quaint. In 1937, Borges was over at the residence of Bioy Casares and his wife, Ocampo, when they started talking about their favorite fantasy stories. Each composed a list, compared it to the others, and then began work on gathering the rights, doing translations, etc. that were necessary for an anthology to exist. In 1940, Antología de la literatura fantástica was released, and later expanded greatly in 1965 and 1976.
The Book of Fantasy is arranged alphabetically, with Borges and Bioy Casares having stories of theirs appear. There is no single definition of "fantasy," nor would the editors have been so inclined. What they did was choose short fictions, some as short as a sentence or two, to appear. Some stories were composed over two thousand years ago, while some were very current (in the revised editions, J.G. Ballard's "The Drowned Giant" appears, for example.) Most are from English or German-speaking countries, although some were from the Middle East, Latin America, and eastern Asia (don't recall any from Africa, in case any are curious).
Some of the names are familiar, such as G.K. Chesterton, or W.W. Jacobs, whose "The Monkey's Paw" is still one of the creepiest short fictions I have ever read (I once read/performed this story for a class of mine two years ago. They really got into it and one even was a bit terrified when the tapping/dragging sounds began).
I was not surprised to see certain themes crop up in those stories that Borges had been discussing earlier in his essay collections. Take for instance this quote from Nathaniel Hawthorne's "Earth's Holocaust":
"My dear sir," said I to the desperate bookworm, "is not Nature better than a book? Is not the human heart deeper than any system of philosophy? Is not life replete with more instruction than past observers have found it possible to write down in maxims? Be of good cheer. The great book of Time is still spread wide open before us; and, if we read it aright, it will be to us a volume of eternal truth." (pp. 133-134)Fits in rather well with some of the observations I quoted from his earlier works, no? In some aspects, this anthology is like reading some of Borges' source material in a very well-designed layout of stories that manage to create and reinforce a mosaic of interpretations for what constitutes "fantasy." There is not a single dud in this collection and most of the works are master stories. Even if you are totally unfamiliar with Borges, The Book of Fantasy is a must-read anthology of great stories.