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As far as horror trilogies go, The first three Alien films make up my favorite, though I don't think any have been as consistently good as The Evil Dead films. The original Frankenstein trilogy with Frankenstein, Bride of and Son of is great as well.
As far as franchises go:
Romero's Dead franchise. Considering that all hail from the same director, some may be disappointing, but NOTLD and DawnOTD alone make it worth the mention.
Dracula. With the 1931, 1958, 1992, and unofficial 1922 version in its repertoire, it stands among the best franchises in general, transcending the genre label.
Dr. Jekyll and Mr. Hyde. Just like most classic horror monsters, its reputation has been slightly spoiled by too many sub par adaptions, but it still has a few noteworthy ones. Such as, good 1920 and 1941 versions, a superb 1931 version, and an Abbott and Costello parody.
Dr. Mabuse. Perhaps not generally well known, but in Germany, Dr. Mabuse is up there with Freddy, Myers and Lecter as far as horror villains go. Not to mention that the franchise has some very good flicks in it.
Lecter. Honeykid may not agree with this, but it's undeniable that the Hannibal Lecter franchise has garnered much success and quite a fan base. Since three of the films follow the same story, I suppose it could be listed under trilogy as well.
Predator. Now that the franchise has more than two standalone films under its belt, I think it's worth a mention.
I suppose the Halloween, NOES, and Friday the 13th films are the most notable horror film franchises, but they're all bloated with too many unnecessary sequels and remakes for me to call them favorites.
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