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The Silence was a religious order created by Tasha Lem. It was an offshoot of the Church forces of the Papal Mainframe. They tried to kill the Doctor to prevent the fruition of a specific prophecy, which stated: "On the fields of Trenzalore, at the fall of the eleventh, when no living creature can speak falsely or fail to answer, a Question will be asked, a question that must never, ever be answered."
This question was "the First Question, the oldest question in the universe, that must never be answered, hidden in plain sight". Dorium Maldovar told the Eleventh Doctor that the question was: "Doctor who?", which was a question the Doctor had been apparently running from his entire life. This question was later revealed to be a code word intended to allow the Time Lords to know that it was safe to return to the universe from the pocket realm that the Doctor had placed them in to end the Time War. The Silence feared that if the Time Lords and Gallifrey returned, the Time War would begin anew and the only way to stop this from occurring was to prevent the Doctor from giving his true name to the Time Lords through a crack in time at Trenzalore.
The Silence was founded by Tasha Lem, Mother Superious of the Papal Mainframe, when she changed the core tenets of the Papal Mainframe to dedicate it to ensuring that "silence would fall" in response to the First Question, sent by the Time Lords to find the correct universe into which they could return. It was believed that the Time War would begin anew if the Time Lords returned to the universe, and so the Church of the Silence was created to prevent the Doctor from answering the First Question and telling the Time Lords that it was safe for them to return. While the main body of the Silence attempted to prevent the Doctor from speaking his name at the Siege of Trenzalore, the Kovarian Chapter broke away to try to stop the Doctor through time travel. (TV: The Time of the Doctor)
Madame Kovarian and Tasha Lem apparently led the Silence. Some members belonged to a race of genetically engineered priests (known as Silents) who used post-hypnotic suggestion to manipulate other species into doing their bidding and to make confessions. All memory of the Silents would also be erased from one's memory after an individual looked away from them. The Silents also employed human or humanoid agents to carry out specific tasks. These agents were often fitted with Eye Drives which allowed them to remember the Silents when they turned away. (TV: A Good Man Goes to War, The Wedding of River Song)
The Silence allied with Madame Kovarian, the Church and the Order of the Headless. They were behind the Battle of Demon's Run and the kidnapping and brainwashing of the infant River Song, then known as Melody Pond. (TV: A Good Man Goes to War) She later made several attempts on the Eleventh Doctor's life. Ultimately, she kissed him with poisoned lipstick. She later broke free of her conditioning and revived him using the remainder of her regenerative energy to save his life. (TV: Let's Kill Hitler)
The Silents were first properly encountered by the Doctor in 1969 America although he had been aware of the existence of the Silence for some time. The Silence had been on Earth since humanity's beginning, secretly manipulating humanity and its development for their own ends.
It was implied they were responsible for the Moon landing because they needed a special spacesuit for Melody Pond for the day when she would kill the Doctor. The Silence's occupation of Earth was broken when the Doctor issued a post-hypnotic command via the Apollo 11 footage to the entire human race. The footage included a recording that Canton Delaware had made of a Silent saying that humans should kill the Silents on sight. (TV: Day of the Moon)
The Silence tracked down River Song to the Luna University, where they kidnapped her and forced her into an astronaut suit as part of their plan to kill the Doctor. (TV: Closing Time) One Silent was present near Lake Silencio in Utah to observe the Doctor, River, Amy and Rory during the events of the Doctor's death. (TV: The Impossible Astronaut) Their plan was foiled when the Doctor convinced the crew of the Teselecta to disguise their ship as him and take his place, tricking the Silence into believing they had killed him at a fixed point in time. The fixed point was actually the Teselecta's destruction and the deception itself. (TV: The Wedding of River Song)
During the Siege of Trenzalore, the event that led to the founding of the Silence, the Kovarian Chapter broke away from the main body of the Silence to try to prevent the Doctor from reaching Trenzalore and potentially speaking his name by going back in time and engineering events to kill him in his past. These events, the destruction of the TARDIS, the creation of the Cracks, and the kidnapping and manipulation of River Song ironically led to the events they were trying to prevent. The Doctor referred to this as the Destiny Trap: "you can't change history if you're a part of it." When the Siege turned into all out war, the Doctor allied himself with the Silence to defend Trenzalore for six hundred years. Together they destroyed or forced to retreat all but the Daleks before being overrun. After the Time Lords changed the future by granting the Doctor a new regeneration cycle, the Crack closed and the Silence's goal was fulfilled as the Dalek forces were destroyed and the Doctor was unable to restore the Time Lords. (TV: The Time of the Doctor)
One Silent was imprisoned in the Judoon prison, along with the Thirteenth Doctor. She stated that she forgot it was there, before being interrupted by Captain Jack Harkness, forgetting it again. The Silent watched the Doctor and Jack escape through the cell walls, using Jack's breakout ball. (TV: Revolution of the Daleks)
When asked by the Doctor why she was in Venice, the Saturnynian Rosanna Calvierri responded, "We ran from the silence". During Isabella's execution, Rosanna's steward recited the line, "And so in memory of the children lost to the silence, the traitor is delivered to the arms of those she betrayed". Following his defeat of Rosanna, when the Doctor was preparing to leave, a mysterious silence fell over Venice. All the people had vanished from a previously busy Venetian market. (TV: The Vampires of Venice)
After attending Amy Pond and Rory Williams' wedding, the Doctor discussed the TARDIS blowing up and warned that "the Silence is still out there" before being distracted by a phone call. (TV: The Big Bang)
When the Eleventh Doctor telepathically connected to a Type 1 TARDIS, it found solace in his memory of the phrase "Silence Will Fall", inspiring it to use the Void to bring order to its perceived chaos of N-Space. (COMIC: The Lost Dimension)
Behind the scenes
Doctor Who: Legacy
In the story of Doctor Who: Legacy, the Eleventh Doctor and his companions find a Silent guarding a time paradox in America on 8 April 1969. The Doctor claims that the Silence should not be here but further in his past, which he attributes to damage to time caused by the Sontarans' interference in the timeline.