In the climactic scene of The Lord of the Rings, the One was destroyed in the Crack of Doom at Orodruin, where it was originally forged, causing the downfall of Sauron and the undoing of all he created with it.
Galadriel tells Frodo that with the destruction of the One, the power of the other surviving Rings, though no longer bound to it, would fail. At the end of the Third Age, the Three were taken to Valinor by their bearers.
Dragons consumed four of the Seven Rings, and after Sauron's return in the Third Age he recaptured three of them. These were almost certainly buried in the rubble of Barad-dûr.
The fate of the Nine was likely the same as that of the remaining Seven. Tolkien wrote that "all of the Nine had come into the possession of Sauron", and that "they perished and became useless when the One was destroyed". This implies (contradicting a statement made by Gandalf at the Council of Elrond) that the Nazgûl no longer physically possessed the Nine. This implication is supported by the fact that the Witch-King's ring was not captured after he was killed. Leaving aside the peril of using it, Aragorn and Gandalf would not have allowed a Ring of Power to be left unattended on the battlefield. However, it is just as likely that the Nazgul's Nine Rings were still worn by each of them, but were automatically destroyed by the unmaking of the One Ring.
The bearers of the Rings of Power in Jackson. From top to bottom: Elves, Dwarves, MenRalph Bakshi's 1978 animated film begins with the forging of the Rings of Power and the events of the Last Alliance's war against Sauron, all portrayed in silhouette against a red background.
Peter Jackson's The Lord of the Rings: The Fellowship of the Ring begins with a similar prologue, though longer and more detailed. The three Elven rings are shown being cast using a cuttlebone mold, an ancient primitive casting technique consistent with the book's description of them as "only essays in the craft before it was full-grown". Additionally, Tolkien illustrators John Howe and Alan Lee, employed as conceptual designers for the films, have cameoes as two of the nine human Ring-bearers (the future Nazgûl).