Is the song just the way you look tonight in the public domain?

No, "The Way You Look Tonight," written by Jerome Kern and Dorothy Fields in 1936, is not in the public domain. As a piece of music created after 1923, it remains under copyright protection. Typically, works published after 1978 are protected for the life of the author plus 70 years, and for works published before 1978, the copyright lasts for 95 years from publication.