I'm partial to stories about ghosts and the supernatural. Supernatural literature goes back centuries; it's a rich source of themes and ideas for modern entertainment. Here are three of my favorite stories-turned-into-movies.
So he put his hand into the well-known nook under the pillow: only, it did not get so far. What he touched was, according to his account, a mouth, with teeth...
M. R. James, "Casting the Runes", in More Ghost Stories (1911)
The opening to "Casting the Runes", quoting three letters of regret to a Mr. Karswell, suggests that it will be an epistolary story, and a great deal of what happens is conveyed in an indirect, old-fashioned style. "This much is in the way of prologue," and "The next scene that does require to be narrated is a conversation..." We discover ourselves in a supernatural murder mystery, a compelling example of Victorian horror. James Hynes, who includes a pastiche of "Casting the Runes" in his collection Publish or Perish, describes James as "author of the the greatest ghost stories in the English language."
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James's story became Night of the Demon, or Curse of the Demon, with Dana Andrews and Peggy Cummins. The 1957 movie is probably best known for its director, Jacques Tourneur, who directed Cat People in 1942. The story is filmed in a much more direct style, and a few details have been filled in (some to good effect, others less so). The tension in the movie is similar to that of the short story: Will all end happily?
"You were too late," the voice said. "You were a minute too late."
Fritz Leiber, Conjure Wife (1943)
In Conjure Wife, Norman Saylor is a young sociology professor at a college in the northeast. Early in the story, Norman discovers that his wife Tansy is a witch. Being a modern sort (for the 1940s, at least) he persuades Tansy to give up her witchcraft, not foreseeing the consequences.

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Leiber's novel became the movie Night of the Eagle (in the UK, retitled Burn, Witch, Burn! for the U.S. market). The setting was moved to England, which works quite well, in the 1960s, also a good move. The movie simplifies away much of the plot, but it retains the same flavor. Witchcraft is just superstition... isn't it? There's a rational explanation for the odd things that are happening. And what's the worst that could happen if we simply disbelieve?
"Died?" I almost shrieked.
She seemed fairly to square herself, plant herself more firmly to utter the wonder of it. "Yes. Mr. Quint is dead."
Henry James, "The Turn of the Screw" (1898)
"The Turn of the Screw" is one of the most famous literary ghost stories in history, though there's been some debate about whether it really is a ghost story. A governess is hired to care for two young children, Miles and Flora, at their country estate. Unexplained figures come and go across the property; the governess must unravel the mystery as well as she can.

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The story became The Innocents (1961), with Deborah Kerr and Michael Redgrave. (Peter Wyngarde, Norman in Burn, Witch, Burn! plays Peter Quint here.) The Innocents is considered a psychological horror movie; it's also considered one of the best movies of its time.
Update:
The Thing cannot be described—there is no language for such abysms of shrieking and immemorial lunacy, such eldritch contradictions of all matter, force, and cosmic order. A mountain walked or stumbled. God!
H.P. Lovecraft, "Call of Cthulhu" (1928)
I forgot to mention H.P. Lovecraft's 1928 story "Call of Cthulhu", which in 2005 was turned into a marvelous short film by the H.P. Lovecraft Historical Society: a black-and-white silent movie, following many of the conventions of 1920 film-making. While Re-Animator, From Beyond, and In the Mouth of Madness are all worth watching, for Lovecraft afficionados, Call of Cthulhu is remarkable, the best of all the adaptations of his work.

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Salon.com
Comments
Totally agree about Blackwood.
I've added an update for Lovecraft, who's an interesting figure in horror--a really terrible writer (imho) with really interesting ideas.
I've recently discovered that the term "Elder Gods", which I really like, is one of August Derleth's creations, not Lovecraft's.
"A man stood upon a railroad bridge in northern Alabama, looking down into the swift water twenty feet below. The man's hands were behind his back, the wrists bound with a cord. A rope closely encircled his neck. It was attached to a stout cross-timber above his head and the slack fell to the level of his knees. Some loose boards laid upon the sleepers supporting the metals of the railway supplied a footing for him and his executioners--two private soldiers of the Federal army, directed by a sergeant who in civil life may have been a deputy sheriff." (An Occurrence at Owl Creek Bridge)
Love Bierce's stories, and I'll always wonder what happened to him in Mexico. Or wherever....
I'm also reminded of E.F. Benson, who is known for his Mapp and Lucia stories, but who also wrote great ghost stories.
D.K was an outstanding actress.Her eyes were special.She gave the impression to always look behind the veil.The story is supernatural.People who have never seen a movie like this,better make sure they watch it in company.
Another good movie is"Wait until dark",starring Audrey Hepburn.It's a crimi.
R
Wait until Dark is a nice suspense film. It's been ages since I've seen it, but I remember Hepburn being pretty convincing as a blind woman and Alan Arkin being pretty convincing as a psycho.
Will found The Innocents at one of his dubious websites. Also, it's at Youtube, if you don't mind that the soundtrack is in another language! I think I'll try reading the Henry James original first, then see if I can control my terror to watch!
Snippy, thanks for reminding me of The Haunting of Hill House. (How could I not have mentioned it?) It's one of the scariest novels I've read. I still get shivers when I read the end of Eleanor and Theodora's shared ordeal. The Haunting captured a lot of it.
"The edge of knife". It's a story about a psychopath.
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You know what I find funny and a little bit pathetic, though? Blurbs on graphic novels (serious comic books) and science fiction novels that favorably compare a genre author, who might possibly be quite good, to Shakespeare. (And it's always Shakespeare. Never So-and-so is is a modern Homer, or Boccaccio, or Cervantes, or Austen.) Give me a break.
The 1945 British horror film was also quite frightening with an evil ventriloquist dummy in one of the sequences stealing the soul of the ventriloquist played by Michael Redgrave.
I read the original story by Fritz Leiber "Conjure Wife" in the pulp magazine "Unknown Worlds" but never saw the film.
I also found three other films noteworthy.
"The Body Snatchers" with Donald Southern
"Alien"
"The Thing" done by John Carpenter which followed closely the original story by John Campbell "Who Goes There?"