In the Totten Foundation’s Victorian mansion in New York City, mild-mannered Professor Hobart Frisbee (Danny Kaye) and his seven fellow academics, among them Professor Magenbruch (Benny Goodman), are writing and recording a comprehensive musical encyclopedia. They have been living cut off from the world for 9 years, living without a radio. Thanks to two window washers (Buck and Bubbles) seeking help with a radio quiz, they discover that there are many forms of popular music—including swing, jive, jump, blues, two-beat Dixie, boogie woogie, and bebop—that they know nothing about. Hobart, the expert on “folk music” goes out and explores the music scene before, during and after hours, inviting all the artists he meets to come to the Foundation.
The professors become entangled in the problems of nightclub singer Honey Swanson (Virginia Mayo). She needs a place to hide out from the police, who want to question her about her gangster boyfriend Tony Crow (Steve Cochran). Tony wants to marry her—because a wife cannot testify against her husband. Honey invites herself into the sheltered household, over the objections of Hobart and Miss Bragg, the housekeeper. While there, she introduces them to the latest music, of which they are completely ignorant, aided by many of the musicians Hobart met the night before. The songs they play include "A Song Is Born", "Daddy-O", "I'm Getting Sentimental Over You", "Flying Home", and "Redskin Rumba".
When Miss Bragg delivers an ultimatum, Honey persuades Hobart to let her stay by telling him she is “wacky” about him and introducing him to “yum yum”, i.e. enthusiastic kissing. Smitten, the innocent Hobart (who graduated from Princeton at age 13) scours the city for an open jewelry store so he can buy an engagement ring. He proposes the next morning. They are interrupted by a phone call from her “Daddy”. Pretending to be her father, Tony easily persuades Hobart to come to Rancocas, New Jersey to be married. Delighted, all the professors join in the elopement. (Honey knocks Miss Bragg out and locks her in the closet.)
A minor accident lands them in an inn near Kingston. There, Honey—who is feeling very guilty and is deeply moved by what she has learned about Hobart—realizes that she has fallen in love with him. Tony and his men arrive and reveal the truth. When Hobart goes to tell her that “Daddy's here,” Miss Bragg arrives with the police. Hobart sends them away. Honey shows him her excuses—a blank sheet of paper.
At the Foundation, the professors refuse Miss Bragg's offer of breakfast. Miss Totten and her lawyer arrive, planning to close down the Foundation. Tony's two men break in and hold everyone at gunpoint. Meanwhile, Honey refuses to marry Tony, even though she will never see Frizzy again. Tony descends on the Foundation with Honey and a very deaf justice of the peace and forces Honey to go through the ceremony by threatening the professors and the assembled musicians. When Hobart learns that Honey is being forced, the hostages join to overwhelm the gunmen. The finale, of course, is not decided by guns but by music, its resonance and reverberation, as, inspired by Joshua Fit the Battle of Jericho, the musicians send a drum crashing on one of the henchmen and a professor pulls the rug out from under the other. Frizzy gives Tony a beating.
Hobart overcomes Honey's objections to their marriage—she feels unworthy—with his own compelling demonstration of “yum yum”.