He recorded it at Columbia (1945), Capitol (1957), and finally for his own record label, Reprise (1963). My favorite is the one he did at Reprise. As with most of his Reprise work, his voice is darker, and the singing jazzier (he could play around with the notes of the songs he'd sung hundreds of times by the sixties). Also, the orchestra is much more lush, and the recording cleaner and crisper.
Probably because his audience had grown accustomed to it as his signature closing song, Sinatra used it for the finale of at least four of his television specials: A Man and His Music (1965), A Man and His Music II (1966), A Man and His Music + Ella + Jobim (1967), Francis Albert Sinatra Does His Thing (1968). Here is each of those: