Not a great deal is known about Moya and her past, except that she was brought into the Peacekeeper fleet and has been used by the Peacekeepers in a genetic experiment and also as a prisoner transport.
Moya had a most unusual birth (according to the novel Dark Side Of The Sun anyway). Her mother gave birth to her near a black hole, but shortly after Moya's mother couldn't escape the black hole. Moya managed to, but had aged differently to the rest of the universe. Moya had only taken a few minutes to escape from the black hole, but because black hole sucks in time, to an outsider it would appear for her to have taken several centuries.
Moya was quite a young Leviathan when her first Pilot was installed, and together they were forced to serve the Peacekeepers who used Moya for a prisoner transport, until Lieutenant Velorik and Captain Crais wished to use her for a secret project. With Moya under anesthetic, the Pilot refused and was killed, and a new Pilot was grafted in place. Moya was tortured until she accepted her new Pilot, whom she eventually formed a relationship with.
Meanwhile, it would seem that Crais's secret project went ahead, apart from Velorik managed to sabotage it. Moya was put back on prisoner transport duty until some of her prisoners (Pa'u Zotoh Zhaan, Ka D'Argo and Rygel) managed to escape and take Moya with them to freedom.
It was not long until the Peacekeeper-implanted Paddak beacon started broadcasting Moya's location though, and Moya landed on a bog-planet to muffle the signal as her crew searched for some anesthetic so they could remove the beacon. However, the gravity made Moya feel her own intense weight, which nearly killed her until the beacon had been removed and Moya managed to free herself.
However, it was not long until more Peacekeeper devices started acting up, and D'Argo accidentally smashed one. This made Moya pregnant as it had been a contraceptive wall, and Moya started turning the DRDs on her own crew, over-protective of her new growing fetus. However, the crew managed to convince her they would not harm her baby.
Moya's pregnancy was an arduous time for her crew, who talked of leaving Moya as she became a less comfortable ship and could possibly lead them into trouble. Moya, hearing their concerns, starbursted to prove herself, but she was not ready and became wedged between dimensions, with three other versions of herself. It was necessary for the crew to go to the three different versions to subsitute for Pilot of Moya so that they could be merged into the original Moya.
Moya's actual birth resulted in a gunship, the product of the Peacekeeper's secret project, who was taken away by Crais. As such, Moya was worried for her son and tried to talk him into stopping, but Moya's son Talyn managed to escape, confused. Later, under the guidance of Crais who treated Talyn well, Talyn returned to Moya's side later. But during Talyn's absence, Moya was judged by her Gods the Builders, who wished to destroy Moya, but Zhaan convinced them to give her a second chance.
However, one of the crew's plans to gain enough money to buy D'Argo's son backfired when the money started eating Moya. The crew suggested partially burning Moya to kill the creatures, which lead to Moya seriously becoming ill. Talyn helped comfort his mother, and the crew found a surgeon to help heal her further.
Moya is not only the crew's ship, but also a member of their crew that they can count on. Her own creature, sometimes she doesn't follow the advice of her Pilot and acts independantly, but always looks out for her son, her Pilot and the rest of her crew.
Moya can be seen as a motherly figure to the rest of the crew, providing them with shelter and sustenance. When she does act independantly she can always be relied on to make the right decision.