Moya was once a captive of the Peacekeepers who sought to control her and other Leviathans through the use of giant control collars placed around the ships to subdue them. Moya and Pilot, with the help of Crichton, Zhaan, and D'Argo, managed to escape.
Moya has become a friend and home to her crew and they have been through many bizarre, disheartening, and exhilarating events together.
It isn't clear how big Moya is, but it seems to take at least a minute or two to walk from one location to another. Even running, it takes at least 30 seconds to get from the command deck to Pilot's den. One novel describes Moya as being the size of Manhattan island, but this may not be canon.
Moya is maintained by DRDs (Diagnostic Repair Drones), small, beetle-like robots.
At the end of Season One, Moya gave birth to a Peacekeeper/Leviathan hybrid gunship that Aeryn Sun named Talyn, after her father.