This hilarious romantic comedy revolves around the misadventures of Dr David Huxley (Cary Grant), 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.
She is a free-spirited young lady, and (unknown to him at first) Mrs. Random's niece. Susan's brother, Mark, has sent her a tame leopard from Brazil named Baby (Nissa) to give to their aunt. Susan thinks David is a zoologist (rather than a paleontologist), and persuades David to go to her country home in Connecticut to help bring up Baby (which includes singing "I Can't Give You Anything But Love" to soothe the leopard). Complications arise as Susan falls in love with David and tries to keep him at her house as long as possible to prevent his marriage.
David finally receives the intercostal clavicle, but Susan's dog George (Asta) steals and buries it. Susan's aunt, Elizabeth Random, arrives. The dowager is unaware of David's identity, since Susan has introduced him as "Mr. Bone". Baby and George run off, and Susan and David mistake a dangerous leopard who has escaped from a nearby circus (also portrayed by Nissa) for Baby.
Cary Grant and Katharine Hepburn in adjacent jail cells
They are jailed by a befuddled town policeman, Constable Slocum (Walter Catlett), for breaking into 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". David fails to convince the constable that Susan makes things up "from motion pictures she's seen". Eventually, Alexander Peabody (George Irving) shows up to verify everyone's identity. Susan (who has sneaked out a window) unwittingly drags the irritated circus leopard into the jail; David saves her, using a chair to shoo the big cat into a cell.
Several weeks later, Susan finds David (who has been jilted by Alice because of her) working on his brontosaurus reconstruction at the museum. After giving him the missing bone (which she found by trailing George), she tells him she has persuaded her aunt to make the large donation.
Against his advice, Susan climbs a tall ladder next to the dinosaur to be closer to him. When the ladder starts swaying from side to side dangerously, she climbs onto the skeleton. Before it collapses, David grabs her hand. Surveying the wreckage of his work, David gives up and admits that he cannot live without her....