Robert Davis (Astaire) is a famous New York dancer in Argentina to play the ponies. He runs out of cash so he goes to the biggest club owner (Adolphe Menjou) to ask for a job. Unfortunately, it turns out that he's a huge asshole and he flat refuses to see Rob. Undaunted, Rob talks to his friend Xavier Cugat, the bandleader, and scores a debut at Menjou's oldest daughter's wedding.
In Shakespearean fashion, Daddy has decreed that his four daughters will get married in order of age, which is crushing for the two youngest, since ice cold Maria (Hayworth), the next in line, keeps steadfastly refusing to fall in love. Rob even tries his luck with her at the reception and gets shut down. He complains unwittingly to her father, who becomes concerned that there might be a problem with his dear girl.
This is where it veers sharply into Weird territory. See, Daddy has a plan. He will manufacture a 'secret admirer' for Maria to fall in love with and then supply some poor sap after a suitable time. So he spends weeks sending her love notes and orchids and, sure enough, she gets curious about her mysterious benefactor. Then he goes on a business trip and completely forgets to continue the orchid delivery. So he comes home, realizes that Maria is depressed, and freaks, frantically ringing for a bellhop to get some orchids and deliver them. Rob intercepts the request and decides he'll deliver the flowers in the hopes of currying favor with the big boss. Maria, waiting by the window, sees him drop them off and recognizes him from the wedding. You can see where this is going.
Daddy is horrified when he finds out, since the last thing he wanted was for Maria to end up with some worthless dancer with a gambling problem. So he comes up with a cunning plan for Rob to break up with his daughter in exchange for a contract in his club.
It's a musical so you can probably fill in the blanks from this point. Hayworth's evening gowns are stunning, of course, even though her expressions are a little on the mugging side. Astaire fell a little flat for me this time, because his character was supposed to be kind of dumb and unsure and I hate seeing him act like that. Adolphe Menjou really stole the show here, despite the creepy aspect of writing love notes to his own daughter. His interactions with his secretary (and his wife's third cousin), Fernando, were hilarious.
If you're a Hayworth fan, it's worth it. If you're an Astaire fan, it's not really.