(covers information from several alternate timelines)
This page contains information regarding Star Trek: Picard, and thus may contain spoilers.
In the anti-time future, in 2395, this ship was operated by Starfleet Medical. This medical ship was commanded by Captain Beverly Picard. The Pasteur was among several hospital ships given special permission by the Klingon Empire to enter former-Romulan space and assist Terrellian plague victims on Romulus.
The ship was destroyed by two Negh'Var warships after the Pasteur crossed the Klingon border to scan the Devron system for a temporal anomaly. However, the survivors of the attack were rescued beforehand by the USS Enterprise-D, under the command of Admiral William T. Riker. Among the Pasteur's crew of that timeline was Ensign Nell Chilton, who died during the Klingon attack. This timeline was replaced by a new one created by Jean-Luc Picard. (TNG: "All Good Things...")
According to the Star Trek Encyclopedia (4th ed., vol. 2, p. 128), Journey's End: The Saga of Star Trek: The Next Generation, and an interview with Michael Okuda on the TNG Season 7 DVD special feature "The Making of 'All Good Things…'", an early pre-production version of the ship's dedication plaque designated the vessel as belonging to the "Hope-class" starship, before being changed to Olympic-class. It would also, however, erroneously list the ship's registry as NCC-58928.