David Huxley (Cary Grant) is a mild-mannered paleontologist. For the past four years, he has been trying to assemble the skeleton of a Brontosaurus but is missing one bone: the "intercostal clavicle". Adding to his stress is his impending marriage to the dour Alice Swallow (Virginia Walker) and the need to impress Elizabeth Random (May Robson), who is considering a million-dollar donation to his museum.
The day before his wedding, David meets Susan Vance (Katharine Hepburn) by chance on a golf course when she plays his ball. She is a free-spirited, somewhat scatterbrained young lady unfettered by logic. These qualities soon embroil David in several frustrating incidents.
Susan's brother Mark has sent her a tame leopard named Baby (Nissa) from Brazil. Its tameness is helped by hearing "I Can't Give You Anything But Love". Susan thinks David is a zoologist, and manipulates him into accompanying her in taking Baby to her farm in Connecticut. Complications arise when Susan falls in love with him and tries to keep him at her house as long as possible, even hiding his clothes, to prevent his imminent marriage.
David's prized intercostal clavicle is delivered, but Susan's aunt's dog George (Skippy) takes it and buries it somewhere. When Susan's aunt arrives, she discovers David in a negligee. To David's dismay, she turns out to be potential donor Elizabeth Random. A second message from Mark makes clear the leopard is for Elizabeth, as she always wanted one. Baby and George run off. The zoo is called to help capture Baby. Susan and David race to find Baby before the zoo and, mistaking a dangerous leopard (also portrayed by Nissa) from a nearby circus for Baby, let it out of its cage.
David and Susan are jailed by a befuddled town policeman, Constable Slocum (Walter Catlett), for acting strangely at the house of Dr. Fritz Lehman (Fritz Feld), where they had cornered the circus leopard. When Slocum does not believe their story, Susan tells him they are members of the "Leopard Gang"; she calls herself "Swingin' Door Susie" and David "Jerry the Nipper". Eventually, Alexander Peabody (George Irving) shows up to verify everyone's identity. Susan, who escaped out a window during a police interview, unwittingly drags the highly irritated circus leopard into the jail. David saves her, using a chair to shoo the big cat into a cell.
Some time later, Susan finds David, who has been jilted by Alice because of her, on a high platform working on his brontosaurus reconstruction at the museum. After showing him the missing bone which she found by trailing George for three days, Susan, against his warnings, climbs a tall ladder next to the dinosaur to be closer to him. She tells David that her aunt has given her the million dollars, and she wants to donate it to the museum, but David is more interested in telling her that the day spent with her was the best day of his life. They profess their love for each other as Susan unconsciously swings the ladder from side to side, and as it sways more and more with each swing Susan and David finally notice the ladder moving and that Susan is in danger. Frightened, she climbs onto the skeleton, causing it to collapse, and David grabs her hand just as she falls. After she dangles for a few seconds, David lifts her onto the platform. After she talks him into forgiving her without him saying a word about anything but halfheartedly complaining about the loss of his years of work putting together the skeleton, David, resigning himself to a future of chaos, hugs and kisses Susan.
