David Huxley is waiting to get a bone he needs for his museum collection. Through a series of strange circumstances, he meets Susan Vance, and the duo have a series of misadventures which include a leopard called Baby.

In this screwball comedy, heiress Susan is determined to catch a stuffy paleontologist and uses her pet leopard, Baby, to help get his attention. The elements of this farce include a yappy terrier who steals and buries an irreplaceable fossilized bone, a pompous big game hunter, a rich old aunt, a jealous fiancée, and a case of mistaken identity involving a second, and vicious, leopard.

When Susan Vance (Katharine Hepburn), a scatterbrained young woman, takes a shine to Dr. David Huxley (Cary Grant), a sober palaeontologist, mayhem ensues. Dr. Huxley, engaged to be married and intensely interested in the arrival of the brontosaurus bone required to complete his project at the museum, is inextricably bound up with Susan's escapades when she finds herself responsible for Baby, a tame leopard shipped to her New York apartment and intended for Susan's aunt in Connecticut. The plot thickens when George the Terrier steals the priceless bone and buries it, Baby escapes, and an untamed leopard escapes from a circus convoy (resulting in a twist on the mistaken identity ploy). Will Susan's love for David be reciprocated? Will David get his bone back? Will Baby be brought up - or brought down?