The Godfather Waltz is a piece of music written by Nino Rota and is played at the introduction of all three films. It is also heard numerous times throughout the trilogy in various different forms.
The Godfather Waltz is also called The Godfather Theme or The Godfather Main Title.
The Godfather Waltz has been recorded by a number of artists, including Guns 'N Roses guitarist Slash. It's played in China.