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While I'm never opposed to it, I tend to think some fandoms really support it more than others do. I've written Oz fanfiction myself, although most of that was prior to ever having heard the term "fanfiction." I think the common term back in the pre-Internet days was "pastiche," although I feel that this is really only appropriate when you're purposely trying to ape the original writer's style. I think the fact that the publishers found other writers to continue the series after L. Frank Baum's death means it always kind of came across as a collaborative effort to me. Even Baum wrote in his introductions about using ideas from readers. Since many of the books are now in the public domain, that means there have been some professionally published fan-written Oz books. On the other hand, I know there's Discworld fanfic, and I can't bring myself to read any of that. It just seems to me that Terry Pratchett's own style is integral to those stories, and stuff written about the same universe by someone else just wouldn't be the same. Then again, maybe that's how I would have felt about Oz books by anyone other than Baum if I'd lived in the early part of the twentieth century. I guess a lot of it is precedent.
As far as TV-based fanfic goes, I really don't read it, but since TV shows are usually collaborative efforts anyway, I don't think I'd have the "not the original writer" objection to it. I'm just not really interested, even if it's based on shows I like. Overall, though, if you like the characters and want them to have other adventures, I don't see any problem with writing them yourself. I'm not entirely sure why it's so popular among fanfic writers to make unlikely characters have sex, but even then I think the important thing is making sure they remain in character. If you're going to make the established characters act like they wouldn't really act in anything other than a parody, why even use those characters?