Astaire . Together, from 1933 to 1939, they made nine musical films at RKO: Flying Down to Rio (1933), The Gay Divorcee (1934), Roberta (1935), Top Hat (1935), Follow the Fleet (1936), Swing Time (1936), Shall We Dance (1937), Carefree (1938), and The Story of Vernon and Irene Castle (1939) ( The