The main story of The Godfather Part III takes place in 1979.
The film begins with Michael Corleone attempting to make his business legitimate and strengthen his relationship with the Catholic Church. Its plot is set roughly two decades after the principal events of The Godfather Part II, placing Michael in an older and more politically exposed phase of his life.
The 1979 setting is important because the story involves the Vatican Bank and the real-world collapse of Banco Ambrosiano, a scandal that became one of the film’s historical inspirations. The movie fictionalizes those events through Michael’s negotiations and alliances.
The third film was released in 1990, so its release year is not the same as the year in which its story is set. That distinction is a frequent source of confusion when discussing the trilogy’s chronology.