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King Arthur was a king of Camelot, one whose exploits were the stuff of legend into the 20th and 21st centuries. It was debated among various sources whether Arthur was a real historical figure or was nothing more than a legend, and, if he existed, whether he was a native of the Doctor's universe or a parallel world connected to the Doctor's England by an interdimensional rigt.
According to some sources, Camelot and its King were part of established history, being foundational to England's causal nexus; both the Master and another Renegade Time Lord tried to meddle with King Arthur's life as part of attempts to unbalance Earth history, and were foiled by the Doctor. (PROSE: The Creation of Camelot, One Fateful Knight) According to other accounts, Camelot and its king existed in another world, only existing as stories in the Seventh Doctor and Ace's native universe. (TV: Battlefield) due to "Merlin's dreams" echoing into the minds of the locals on Earth through the interdimensional rift which connected the two worlds in Carbury. (PROSE: Legends of Camelot) Yet another account claimed that Arthurian stories were largely correct, but occurred in "a time of myth and legend" which was hard to reconcile with the established literary history of the concept as it was known in the 21st century. (PROSE: A Honeycomb of Souls)
At any rate, the human Arthur was actually a "reflection" of the Guardian of Might, an interdimensional entity from before the universe, much as the various versions of "Morgan" or "Morgaine" he faced were reflections of the Guardian of Magic. The story of Camelot was influenced by the dreams of these two entities as part of the gambit of the original Merlin, the Champion of Neutrality, to prevent the two Guardians' battle from destroying the universe. (PROSE: Legends of Camelot)
The Eighth Doctor once landed with his TARDIS in a castle which turned out to be Camelot, where he was asked to help deliver the duchess Igerna's baby. When she named him "Arthur", the Doctor realised what events he'd stumbled into; remembering his encounter with Morgaine in his seventh life and the destiny that he must fulfill, the Doctor began calling himself "Merlin" and stuck around Camelot. (PROSE: One Fateful Knight) Although it was generally believed that Arthur's father was Uther Pendragon, the previous king and Igerna's husband at the time of birth, (PROSE: The Creation of Camelot, Legends of Camelot, One Fateful Knight) she claimed that the baby had been conceived before her marriage to Arthur with her previous husband, the duke Gorlois. This was the cause of much gossip in Camelot, as the very night she claimed that Gorlois had visited her one last time was the night it was generally believed that Gorlois had been slain in battle. (PROSE: One Fateful Knight)
Merlin secreted the baby Arthur away to be raised by Sir Ector as his adopted father, with Sir Kay as his adopted brother. Arthur sometimes acted as Kay's squire; he grew up in relative obscurity until he was old enough to take his place in his fated destiny, which coincided with the Tenth Doctor and Donna Noble's arrival in Camelot. (PROSE: Legends of Camelot)
Ascent to the throne
According to one source, the Doctor pulled the sword from the stone himself and handed it to Arthur, instead of leaving Arthur to do it himself as legend dictated. The Doctor was therefore briefly made King of England, but he abdicated after a day in Arthur's favour. The Twelfth Doctor refused to take Clara Oswald to meet Arthur as he considered this incident a bad time. (PROSE: Silhouette) However, according to another account, while the Doctor was present at this event in his tenth incarnation, it was indeed the young Arthur who pulled the sword from the stone. The Doctor, going by Merlin, then took it from his hands and formally crowned him King. (PROSE: Legends of Camelot)
According to yet another account, Arthur became the king of Camelot after the death of his father, Uther, without special fanfare or troubles. The Merlin of Camelot's court around that time was a certain Blaise, but after he died, he was replaced by a mysterious individual who was actually the Master. (PROSE: The Creation of Camelot)
The Master exposed
A year after the Master arrived, Arthur's son Mordred was born to his half-sister Morgan at Lothian. Arthur sent the Master to kill Mordred. He was unaware that the Master had instead helped Morgan in raising him. The Master planned for Mordred to kill Arthur at the battle of Camlan so Britain could never become one country. The Master pretended to be an enemy of Morgan to fool Arthur.
When the Fifth Doctor and Tegan arrived, they were put into a cell by the guards. Eventually, Arthur came to meet them and set them free. When the Doctor told Arthur about his TARDIS, Arthur told the Doctor that Merlin had mentioned such a thing.
The Doctor realised that Merlin could only be the Master in disguise. Arthur sent for the Merlin . The Doctor and Tegan soon recognised him. The Master tried to convince Arthur that he was no threat and he still served him. Arthur, convinced by what the Doctor told him, had the Master sent back to his room and put on trial. However, he was unaware that the Master's room was actually his TARDIS.
With the Knights of the Table Round
As King of the Knights of the Round Table, Arthur continued to take Merlin's advice and sent his knights on various quests defeating otherworldly threats. After Agravaine prevented the Morch'aliach entity from being truly slain through his recklessness, Arthur and Merlin "chewed him out" together. (PROSE: A Honeycomb of Souls)
Together with his knights and Merlin, Arthur battled the forces of Mordred and Morgaine, but was killed at the Battle of Camlann. Merlin placed Arthur's body and Excalibur in a dimension-crossing spaceship and hid it at the bottom of Lake Vortigern, near Carbury, in rural England. (TV: Battlefield) Accounts differed on whether the Merlin who did this, who was a future Doctor relative to the Seventh Doctor, was the Eighth Doctor (PROSE: One Fateful Knight) or a different incarnation further off in his future. (PROSE: Battlefield) Legends sprang up that Arthur had survived the battle, albeit badly wounded, and would reawaken at the time of England's greatest need. In 1997, the Seventh Doctor and Ace discovered Arthur's spaceship and retrieved Excalibur. They discovered that the stories of Arthur's survival were just propaganda and that the King really was dead after all. (TV: Battlefield)
In the 21st century, when Morch'aliach became a threat again, a time-travelling Ganeida helped a temporally-displaced Bedevere, Origin Zero and the Preternatural Research Bureau destroy the beast by using Bedevere's link to Camelot and the entity's ability to summon the souls of the dead against Morch'aliach; speaking a story into existence, the wizard was able to summon blurry ghosts of King Arthur and his entire army of Knights of the Round Table, who charged the entity. This nearly killed it, causing its pocket reality to flicker out; reduced to the size and shape of a mere worm, it was easily finished off by Ganeida. (PROSE: A Honeycomb of Souls)
Another version of Arthur existed in the distant future. (COMIC: The Neutron Knights) In Roma I he was known as Artorius, and like Concuthasta, he existed in the history of Britannia in all of the Known Worlds. (PROSE: Warlords of Utopia)
Behind the scenes
Stephen Fry's unproduced script, The 1920s, was based on the Arthurian legend of Sir Gawain. The Fisher King in Before the Flood is named after the eponymous Fisher King in Arthurian legend, who, like the Doctor Who character, is in need of aid to be restored.