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A Resurrection Ship is a large vessel specifically dedicated to the "resurrection" of humanoid Cylons who are killed. Normally, upon death, their consciousnesses are downloaded into new identical bodies on the Cylon homeworld, but for humanoid Cylons located farther and farther away from home - such as those hidden within the human refugee fleet - this is not possible, as the download range of a humanoid Cylon is limited.
For this reason, when the Cylons began pursuing the Fleet, they incorporated a Resurrection Ship at the center of their pursuit force, to provide their agents with a fallback position; if they were killed, their consciousnesses could be downloaded to new bodies without concern. Without that ship, a killed Cylon would truly die, as their consciousnesses also would be cut off.
Based on testimony provided by Gina Inviere and on reconnaissance photos taken by Starbuck (using the Blackbird), such a ship contains multiple copies of Cylon models, along with all the apparatus necessary for resurrection (Resurrection Ship, Part I).
In "The Plan" and "No Exit", it's shown that the Resurrection Ships secretly carry at least one new body for each of the Final Five. Only the Cavils know about them, having put them there before the Fall of the Twelve Colonies in the hopes that the Final Five would die in the attacks and download into the new bodies. Presumably, they kept new bodies onboard afterwards in case one of the Five died, as Ellen Tigh eventually did.
Pursuing the Fleet
When Pegasus happened upon a Resurrection Ship, later found to be pursuing the Fleet, they discover the ship was escorted by two basestars and approximately a dozen support vessels. It is assumed that the ships were following the Fleet since the exodus from the Colonies. Pegasus located this ship and its escorts sometime after the Fall of the Scorpion Fleet Shipyards and in turn used the Cylon fleet's movements to locate Galactica.
A copy of D'Anna Biers on Caprica tells of two Raiders from this Cylon fleet that are destroyed while relaying a documentary on Galactica's crew back to what she calls "the fleet," presumably the same Cylon fleet guarding this ship (Final Cut).
Prior to Thrace's recon flight, the function and configuration of the Resurrection Ships were a complete mystery. Admiral Cain speculates that it is a command and control ship, while Adama suggests that it might be a Raider factory. Adama's suggestion would actually turn out to be quite close to the truth. The two ships begin planning an offensive strike on it, but tension between Cain and Adama quickly escalates and threatens to derail the operation (Pegasus). The standoff only ends after Thrace's arrival with data regarding the unknown ship.
In a joint operation by both Galactica and Pegasus, the Resurrection Ship following the Fleet is disabled and soon destroyed (Resurrection Ship, Part II) using a number of Colonial civilian transports left to bait the basestar's Raiders away from the Cylon Fleet. The fighters are engaged and kept occupied by the Vipers protecting the "helpless" Colonial ships.
With the Cylon fleet divided, the two battlestars attack the two basestars, their offensive power greatly diminished by the lack of their Raider forces. As the Resurrection Ship attempts to flee, the Blackbird, piloted by Lieutenant Lee Adama, slips within the ship's superstructure and disables the ship's FTL drive. As the two escort basestars are hacked to pieces by the battlestars' gun batteries, the Resurrection Ship, without any defensive support or offensive guns of its own, is quickly destroyed by another Viper squadron who fire their guns through the windows apparently hitting vital systems as it causes a chain reaction which destroys the Resurrection Ship.
After its destruction, the imprisoned Cylon Sharon Valerii reveals to Thrace that the Resurrection Ship had also served to resurrect the consciousnesses of destroyed Cylon Raiders. The loss of the ship forces the Cylons to change their attack tactics from direct frontal assaults on the Fleet en masse to more cautious hit-and-run Raider strikes and coy traps. However, by the time the two Cavils are executed, there is apparently another Resurrection Ship nearby as the Cavil on Galactica tells the Caprica copy that there is one and they will be resurrected. This Cavil may not be aware of the destruction, but this is unlikely as Gina was able to tell and it was likely announced as it is a major victory. If there was a Resurrection Ship, the two Cavils were likely reborn in new bodies, although the Galactica Cavil planned to box the Caprica copy. (The Plan) Despite this, Gina was apparently not resurrected, but the Fleet may have been out of range of this second Resurrection Ship at that point. (Lay Down Your Burdens, Part II)
In the Cylon fleet leaving New Caprica during the Second Exodus, a second Resurrection Ship's existence is confirmed. The Cylons have assigned at least five basestars to guard it as opposed to only two for the previous ship (Collaborators).
When Ellen Tigh is killed by her husband, she downloads into a new body on a Resurrection Ship and is subsequently held captive by Cavil on a basestar until her escape (No Exit).
When a basestar falls prey to a deadly disease, the Resurrection Ship is ordered to jump to a distance outside the downloading area of the ship in order to halt the spread of the disease, thereby resulting in the permanent deaths of the basestar's entire contingent of biological Cylons (Torn).
Later on, this Resurrection Ship nearly falls victim to a plan formulated by Galactica to kill their Cylon prisoners infected with the virus so that the it would spread into the Resurrection Ship, presumably resulting in the infection and death of a great majority, if not the entire, Cylon race (A Measure of Salvation).
After having strange dreams which she believes to be a message from God, a Three orders a Cylon Centurion to execute her, which results in her being reborn in the Resurrection Ship that is traveling with the Cylon fleet. This is the first instance in which the interior of a Resurrection Ship is seen clearly, and it proves to be nearly identical in design to the basestars, with the noticeable difference that it does not have the oscillating strip of light along the walls as seen on the basestars (Hero).
Later, as part of her plan to rescue her daughter, Sharon Agathon has her husband shoot and kill her. This causes her to download into a new body on the nearby Resurrection Ship. Caprica-Six oversees her resurrection and is sympathetic enough to take her to her daughter and ultimately helps to rescue her (Rapture).
Cylon Civil War
Following the detection of the Final Five during the Battle of the Ionian Nebula, a Number One, who was shot by a rival Cylon faction (Six of One), successfully downloads into a new body. Subsequently, as part of a trap for Natalie's faction, the Resurrection Ship is prevented from jumping with the Cylon fleet in order to permanently kill Natalie and her followers (The Ties That Bind). As a result, two of the surviving rebels who die later - a Six executed for killing Jean Barolay, and an Eight mortally wounded by a Centurion - won't be able to resurrect because the rebels don't have access to a Resurrection Ship (Faith). Also Natalie's death is final when she dies of gunshot wounds inflicted by Athena (Sine Qua Non). During the Civil War, all of the Resurrection Ships remain in Cavil's control.
In a massive battle the Cylon rebels and the Colonial Viper pilots launch an attack against the Cylon Resurrection Hub. This Hub controls all of the Resurrection Ships in existence, making it vital for their functioning. The combined force succeeds in destroying the Resurrection Hub and with it all the Resurrection Ships' downloading capabilities, thus making all Cylons' deaths permanent from then on (The Hub). Thanks to the destruction of the Resurrection Hub and the disabling of the Resurrection Ships as a result, all of Cavil's faction dies, including Boomer and Cavil himself when Galactica destroys The Colony during the Battle of The Colony. Also, Samuel Anders and Tory Foster's deaths are permanant.