On the eve of his wedding, Dr. David Huxley, a dedicated paleontologist at the Stuyvesant Museum of Natural History, is sent by his fiancée and assistant, Alice Swallow, to play golf with Alexander Peabody, the lawyer for Mrs. Carleton Random, a potential million-dollar donor to the museum. At the golf course, flighty heiress Susan Vance plays David's ball instead of her own and then, mistaking his car for hers, drives off with him clinging to his runningboard. That night while hunting for Peabody at an exclusive restaurant, David again encounters Susan, who causes him to slip on his top hat, embarrass himself in front of psychologist Dr. Fritz Lehman, tear his jacket and split the back of her gown. The next morning, Susan telephones David, who is preparing to meet Alice with his new possession, a rare brontasaurus fossil, and begs him to help her with her new possession, "Baby," a tamed leopard that her brother has shipped to her from Brazil. David, however, refuses to get involved with Baby until he hears Susan's phony cries of distress over the telephone. After rushing to her apartment, David finds Susan unmaimed, and Baby yearning to hear his favorite record, "I Can't Give You Anything But Love." Disgusted by Susan's antics, David marches out of the apartment, but is followed down the street by both Susan and an unleashed Baby. Thus cornered, David finally agrees to help Susan take Baby to her aunt Elizabeth's home in Connecticut, but admonishes her that he has to return to the city to marry Alice by nightfall. While driving on the road to Aunt Elizabeth's, a distracted Susan rams into a truck carrying a load of fowl, and its cargo spills out and is devoured by Baby. Later, while David is buying raw meat for Baby in a small town store, Susan is forced to steal a stranger's car whose back seat the leopard has suddenly occupied. Finally arriving in Connecticut, David, who has donned Susan's dressing gown because Susan has sent his feather-encrusted clothes to the cleaners, runs into the befuddled, suspicious Aunt Elizabeth, whose married name is Mrs. Carleton Random. Because David has asked her not to reveal his full name to Elizabeth, Susan tells her aunt that David's last name is "Bone" and that he is a big game hunter who has suffered a nervous breakdown. At the same time, Elizabeth's dog George steals David's bone and buries it on the vast estate. While David frantically follows George around the wooded estate in an attempt to discover the whereabouts of his fossil, Susan confesses to Elizabeth that she is in love with David and plans to marry him. Unwilling to leave Elizabeth's without his fossil, David joins Susan, Elizabeth and Major Horace Applegate, a true big game hunter, for dinner. While David carefully watches George from the table, Mr. Gogarty, a heavy-drinking family servant, accidentally releases Baby from his makeshift cage in the garage. Alerted by Gogarty's screams, Susan orders David to telephone the local zoo, but then tells him to cancel his request for help after she learns that her brother intended Baby as a gift for Elizabeth. On the estate grounds, Susan and David search for Baby, harmonizing "I Can't Give You Anything But Love" as a lure, but mistake a caged, vicious circus leopard, which is being trucked to Bridgeport, for their tame animal. After Susan surreptitiously releases the other leopard from the stalled truck, it escapes into the woods and ends up on the roof of Dr. Lehman's house, where she and David attempt to coax it down. Lehman comes to his front door and, seeing only Susan, drags her into his house, convinced that she is deranged. Constable Slocum then arrives on the scene, spots David slinking around the house and arrests him for voyeurism. At the jail, Slocum refuses to believe Susan's and David's stories and arrests both Elizabeth and Applegate when they come to bail out Susan because he is sure they are only impersonating his wealthy constituents. Unable to persuade the dim-witted Slocum of her true dilemma, Susan changes her tactics and pretends to be "Swinging Door Susie," a gangster's moll. Eventually, Peabody shows up to verify everyone's identity, and after Baby and George stroll into the station, Susan, who has snuck out of a window, unwittingly captures the circus leopard. A few weeks later, Susan finds David, who has been jilted by Alice, working on his brontasaurus reconstruction at the museum. After presenting him with his bone, which George finally had returned, Susan informs David that she is donating a million dollars that Elizabeth has given to her to the museum. Then while perched on a tall ladder that scales the dinosaur, she extracts a confession of love from David. Although the excited Susan causes the one-of-a-kind reconstruction to collapse in a heap, David laughs at his misfortune and embraces his bride-to-be.