The Flight of the Hippogriff was a pivotal event that took place on the evening of 6 June 1994, in which Harry Potter and Hermione Granger used a Time-Turner to successfully rescue the Hippogriff Buckbeak from execution and save Sirius Black from receiving the Dementor's Kiss. The event involved flying Buckbeak to the West Tower of Hogwarts Castle, where Sirius was being held, and facilitating his escape from the castle grounds. This act saved the lives of two innocent beings and allowed Sirius Black to go on the run as a fugitive.
The events leading to the flight were twofold, involving the separate plights of Buckbeak and Sirius Black.
Realizing both Sirius and Buckbeak were innocent victims facing imminent death, Albus Dumbledore cryptically advised Hermione to use her Ministry of Magic-approved Time-Turner to go back three hours, hinting that she could save “more than one innocent life.”
Following Dumbledore's advice, Harry and Hermione orchestrated the rescue with precise timing.
The Flight of the Hippogriff was a resounding success with profound consequences for the remainder of the series.