The fantasy short story goes back to Lord Dunsany, whose first stories came out in the early years of the 20th century. Well, it's not really as simple as that, of course. Fantasy has been around since the dawn of storytelling, and so have short stories. Still, Dunsany was really the first writer of modern fantasy who excelled at the short story, but most of his work consists of stand-alone stories, give or take the odd sequel or recurring character.
It was in the U.S. pulp magazines that the fantasy short story really came of age, in the hands of authors like Robert E. Howard, Clark Ashton Smith, H.P. Lovecraft (yes, he wrote fantasy as well as horror) and their successors, and it was here the series came into its own. Some were only loosely connected by setting, such as Smith's tales of Zothique, Hyperborea, Xiccarph and many others. Other authors focused on a specific character, or pair of characters: Howard's Conan, Fritz Leiber's Fafhrd and the Gray Mouser, C.L. Moore's Jirel of Joiry, Fletcher Pratt and L. Sprague de Camp's Harold Shea, and countless others.
The tradition survived well into the 60s and 70s, though by this time there was more of an impetus to write novels. Both Michael Moorcock's Elric and Karl Edward Wagner's Kane featured both in novels and short stories, for instance. The process has developed, and novels are now the default for any fantasy author wanting to write something ongoing, while the series of shorter stories is now predominantly the domain of TV.
Most short stories now are stand-alone, but the series isn't entirely dead. I love the format, and I've developed several of my own. There are the stories relating moments in the Traveller's millennia-long journeys. There are the tales of Eltava's life, many of them overlapping with the Traveller's series. There are the wanderings (both in a physical and a moral sense) of teenage sorcerers Karaghr and Failiu, a lighter series perhaps more in Leiber's tradition. And then there's the completely unrelated series about spoof hard-boiled detective Sam Nemesis.
So what's the attraction of the series, and what are its advantages and disadvantages?
I suppose the most obvious disadvantage must be the risk of getting repetitive, of falling into a predictable formula for every story. On the other hand, this pitfall isn't restricted to a series: an unimaginative author can easily recycle the same plot and characters over and over, even when all the names are changed.
There's also the practical issue that, when the stories are published occasionally, here and there, it's necessary effectively to introduce the main character and the concept for each story. Back in the days of the pulp magazines, the Conan stories, for instance, (almost) always appeared in Weird Tales, and Fafhrd and the Gray Mouser in Unknown. It meant that the author could assume that a large proportion of the readers were familiar with the series, in the same way that a TV series doesn't need to introduce first principles every episode.
That isn't true with current short story magazines, whether print or online: readers don't generally want to see the same authors and the same characters coming up issue after issue, and writers of a series have to hawk the stories around, getting into one publication here and another there. The seven published stories in my Eltava series, for example, have appeared in six different magazines.
On the other hand, a series gives the opportunity to develop a character or a setting in a way that's usually only possible in a novel. Even if the readers aren't following it consistently and consecutively, as with a TV series, the author is able to gain a deep knowledge of the character and the settings in which he or she appears, if those are constant, and that can translate as a level of confidence and reality that will come over even to the casual reader. It's the same as how, for example, a new viewer of Doctor Who can often feel the weight and significance of the show's fifty years behind what they watch, without having to understand everything that's gone before.
Unlike a novel series (though there are exceptions even with those) a short story series doesn't have to come in sequence, especially when the stories are being published here and there. Indeed, many of the classic series weren't written with any strong sense of sequence at all. The Conan stories have a vague sense of Conan getting older, and they've subsequently been arranged by fans into a firm order, but for the most part they "reset to default" for the beginning of each story.
The original seven stories of Fafhrd and the Mouser, written for Unknown, follow an approximate sequence, largely governed by the pair's four-story-long excursion to the western continent and back to Lankhmar, and the stories Leiber wrote from the 70s onward follow an order, but the stories in between are fairly random. When the whole series so far was gathered into five volumes in the 60s, two of the three stories in volume one were among the last to be written.
On the other hand, a series with a more logical progression, such as the Harold Shea stories, might be written and published in strict sequence. The two approaches can be seen on TV, too. Many older shows, such as the 60s classic The Avengers, were done very much on a reset to default system — nothing that happened in a given story was carried over to the next episode, and they can be comfortably watched in any order at all.
To a slightly lesser extent, this is also true of the original Star Trek, and indeed there are different versions of the correct order for these stories. By the later versions of Star Trek, though, from the late 80s onward, this had changed. Characters changed and developed in a logical way, and the events surrounding them developed too. The first couple of years of The Next Generation could just about be watched out of sequence, though not comfortably, but Voyager couldn't at all.
I've used the full range of options for my various series. The Sam Nemesis tales are completely reset to default — they take place in an undefined mythical setting, where nothing significantly changes. At the other extreme, the stories I've written so far about Kari and Fai (three published, and I'm working on the fourth) follow a logical sequence of events, and to some extent show the characters taking forward what happens in one story to the next.
The Traveller and Eltava are both written completely out of order, especially the Traveller, whose stories might be set thousands of years and thousands of miles apart. In the case of Eltava, I started with the most obvious part of her life to write about (her twenties) and have since expanded backwards and forwards — so far from early teens to late sixties.
The advantage of this is the capacity to explore the character thematically, rather than lineally. A story about the Traveller at the age of four thousand might inspire something he does as a spring chicken of a few hundred, as much as vice versa. I used this approach too in the novel At An Uncertain Hour, but most novels don't have the capacity to explore this kind of thematic development.
Even with those series that are written, the aim is to publish them, carefully arranged in sequence, in a book (of whatever format). An example, a few years back, is the excellent The Servant of the Mantichore by Michael Erhart, which falls halfway between a story collection and a novel. Of course, I'd love to have my various series in book form, but in some ways that's a distraction from the nature of the series: to explore the characters and settings in a form that can grow in strength by being taken together, but is ultimately designed to be read one story at a time.