Screwball Comedy

The Screwball Comedy has a pretty precise definition: a comedy film—usually in black and white, although some were made in color—in which an uptight, repressed, or otherwise stiff character gets broken out of his or her shell by being romantically pursued by a Cloudcuckoolander (or a similar character type). It does not just mean "zany comedy." The Producers, say, is not a screwball comedy, although it is screwy, ballsy, and very funny. It is characterized by fast-paced repartee, farcical situations, escapist themes, and plot lines involving courtship and marriage and showing the struggle between economic classes.

In other words, a Parody of a Romantic Comedy.

Classic screwball comedy examples include (period 1934-1944)

The Awful Truth
Bachelor Mother
Ball of Fire
Bringing Up Baby
Dinner At Eight
Easy Living
Holiday
It Happened One Night
Its a Wonderful World (the film - page at the moment redirects to The World Ends With You)
It Started With Eve
His Girl Friday (A remake of the play/movie The Front Page)
The Lady Eve
Libeled Lady
Midnight
Mr. Deeds Goes to Town
My Favorite Wife
My Man Godfrey
Nothing Sacred
The Palm Beach Story
The Philadelphia Story
To Be or Not to Be
Top Hat
Topper, followed by two sequels. Based on two novels by Thorne Smith, who also wrote the book on which I Married A Witch is based.
Twentieth Century
You Can't Take It with You

Later and modern examples of screwball comedy include:

I Was a Male War Bride
What's Up, Doc?: Peter Bogdanovich's Homage to the genre
Switching Channels: A remake of His Girl Friday (which as noted above was a remake of The Front Page).
The Hudsucker Proxy: Another homage, written and directed by The Coen Brothers
Miss Pettigrew Lives for a Day: A modern Pastiche of the genre
Arthur is about equal parts PG Wodehouse pastiche and screwball pastiche.
Oscar
Date Night
After Hours and Something Wild can be seen as darkly postmodern '80s variations of the genre.
The Runaway Bride
Conversely, the 1928 silent Marion Davies comedy The Patsy can be regarded as a sort of very early prototype for the genre.
Ticktock, a horror novel by Dean Koontz, is deliberately written as a Screwball Comedy.
Dharma and Greg
House Sitter
Harold and Kumar Go To White Castle is a non-romantic version, in which uptight, nervous Harold gets broken out of his shell by laid-back Kumar. And there's a big cat and everything.