The USS Odyssey (NCC-81302) was an Allegiance-class Federation starship launched in 2384. She was one of the first vessels constructed at the Regeneration Shipyards near Ba'ku in former Briar Patch. Equipped with a Romulan cloaking device and a prototype quantum slipstream drive, Odyssey was sent on a mission to the Andromeda Galaxy to stop the Archein invasion of the Milky Way. She was lost and presumed destroyed. (Star Trek: Odyssey: "Iliad")
In fact the Odyssey was trapped in the Andromeda when its slipstream drive was disabled. The captain and senior staff being dead, Lieutenant Commander Ro Nevin assumed command, with their Romulan observer, subcommander T'Lorra as first officer. The crew of the Odyssey spent a couple of years evading the Archein fleet and forming alliances with their enemies, such as the Kelvans.
They discovered that the Kelvan civilization had depended on Omega molecule technology, and that unknown to them this had created the singularity-formation zones that was making their empire and the Archeins' uninhabitable, precipitating a war of attrition between the Archein and Kelvans and prompting the Archein invasion of the Beta Quadrant. ("The Lotus Eaters")
Ro was taken by the wormhole aliens (whom he knew as the Bajoran prophets and the Archeins called the Great Ones) for six months, during which T'Lorra assumed command. During this time they developed a technology for repairing the spacetime shear caused by the Omega molecules, reversing the creation of singularities. Ro was restored and the Odyssey made its return to the Milky Way, only to discover that experimentation with Omega molecules had caused subspace damage which had isolated Risa from the rest of the galaxy and was driving the Romulan homeworlds' star to go supernova. They were able to repair the damage and go home at last, after nearly three years in Andromeda.
After three months of extensive repairs in spacedock, and extended shore leave for her crew, the Odyssey was relaunched under Captain Ro's command. Subcommander T'Lorra continued to serve as executive officer, her liaison position continuing at her request. Much of Ro's crew returned to the Odyssey, with the exception of Ensign Joshua Gillen who decided to leave starship life behind. Gillen was replaced as chief engineer by Commander Corey Aster. ("Tossed Upon the Shore")