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Monster-Thriller--the original The Haunting from the 1960s.
Mystery--Hitchcock's Rope, also Rear Window.
Romantic comedy--When Harry Met Sally
War movie--this is a bit tougher but a good candidate would be The Hill, which takes place inside a British stockade in the North African desert during World War II with Ossie Davis and Sean Connery. My one hesitation is that there might be a couple of guards with rifles seen at the start of the film when they're delivering a truckload of prisoners, but I can't recall for sure. However, with that possible exception, the rest of the film is completely gunless as both the prisoners and guards inside the camp are unarmed, except for the nightsticks the guards carry. Other than that, perhaps 5 Fingers, with James Mason as a WWII spy, which I remember as essentially gunless although again there could be soldiers standing guard in the background like statutes with slinged rifles.
Western--now this is really hard although I know there must be one out there somewhere. But the best I can do on this is an odd fact about the John Wayne classic The Angel and the Badman. Oh, there are certainly guns and certainly shooting--as the opening credits are rolling you see in the background 4-5 hip-level shots of Wayne (or maybe his standin since you don't see the face) firing his pistol. But after that opening and through the rest of the movie, John Wayne never fires a single shot! There's a scene where he bluffs bad guy Bruce Cabot with an empty pistol, but later he uses clubs instead of guns to take a cattle herd back from rustlers. And he's ready to kill Cabot and his gunman near the end of the picture when his Quaker girlfriend prevents the shootout, leaving it to Harry Carey to save her gunfighter. I just always liked it that Wayne had the guts to play a gunfighter without ever shooting anyone.
Other than that, any "gunless" Western would have to be something about pre-Columbian Indians like that thing years ago with Yul Brynner and George Chakarus (OK so I can't spell) playing heads of opposing tribes with stone-age weapons. A Man Called Horse is almost gunless except for Harris's trapper guides shooting up the camp at the start of the film. (But not that film Mel Gibson directed as in the end we see Spanish soldiers headed for the shore.
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let the fun begin
my kung fu is weak, i know i should have made it more than 5 movies 
