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The Monks were an alien race who used a virtual simulation of life on Earth in order to plan an invasion. Virtual constructs included simulations of the Twelfth Doctor, Nardole and Bill Potts. (TV: Extremis)
They were also able to see into the future through their simulation, and vowed to "protect" the Earth from the devastation they foresaw would occur in one year. However, they required genuine consent from people with power, and were able to sense if their consent was true or they were acting out of fear or strategy. If their consent was not pure, the person would immediately be disintegrated. (TV: The Pyramid at the End of the World)
It was clear that the Monks showed no mercy for the human race in any way, and seeked to control the Earth and humanity itself. They had no problems instantly killing people, and on multiple occasions threatened the Doctor himself.
There was no clear hierarchy within the species, but it appeared that few Monks spoke while the rest remained silent. There would also be one Monk whose job would be to feed the invaded population falsehoods about their personal history. (TV: The Lie of the Land)
The Monks chose their particular appearance on Earth to resemble human corpses. (TV: The Pyramid at the End of the World) As a result, they had greyish-blue skin, pointed ears and long fingers. They typically wore robes and cowls with various shades of red, hence the name given to them. (TV: Extremis) They were apparently shapeshifters, capable of choosing external appearance at will. (TV: The Pyramid at the End of the World)
They spoke with a snarling, whispering voice, and their mouths did not move in sync with their voices. They typically moved quite slowly, and walked with their heads down and their hands together.
They were able to sense the purity of one's consent simply by placing their hand over one's chest. If their consent was not pure, then the Monks were able to disintegrate them. (TV: The Pyramid at the End of the World)
They were able to shoot out lightning from their hands, as well as make shields from this energy. However, they weren't invulnerable and gunfire that penetrated their shields could kill them. (TV: The Lie of the Land)
The Monks had extremely advanced technology. One of their notable achievements was a computer simulation of the entire planet Earth in which they ran scenarios and practiced and perfected their invasion strategy, and also in order to assess the strengths and weaknesses of humanity. However, their simulation was too perfect. The simulations of real people that were part of the program were capable of gaining sentience. Once they realised the truth that they were nothing more than a program, they would choose to kill themselves rather than be constantly used and abused by the Monks. (TV: Extremis)
They managed to make a 5,000-year-old pyramid appear out of nowhere between the three largest armies in the world by the 21st century - America, Russia and China. This pyramid, called the Cathedral (TV: The Lie of the Land) was their base of operations on Earth and housed technology which enabled them to see into the future of their virtual Earth, as well as their spaceship on and off Earth..
They were also able to use threads from their simulation of Earth to see into the future, and predicted a World War III-like event where all human life would be wiped out in one year. They were able to spy on the human race through CCTV cameras, and were also able to hack into the technology of normal citizens of Earth, such as mobile phones or watches, and change the time to issue warnings about the safety of the Earth.
It was later revealed the time changed in accordance with activities that unfurled within a science lab in Yorkshire when a scientist accidentally misplaced a decimal point and inserted 118.9 instead of 11.89 into the computer, killing their research immediately. The Doctor said that this event was going to 'blow up in their faces' and was the reason for the state of Earth seen by the Monks in one year. Also, the Monks could hijack vehicles and other technology and appeared to be able to teleport themselves into areas such as planes. (TV: The Pyramid at the End of the World, The Lie of the Land)
After the Monks get the consent they wish for to invade, they erect statues in most major areas in order to broadcast lies about the invade planet's past. In The Cathedral, a Monk would be wired to what the Doctor called "Fake News Central" which they would broadcast the lies using the brainwaves of whoever gave their consent to invade the world. (TV: The Lie of the Land)
For millennia, the Monks invade planets by studying them through the Shadow World, finding someone in power to ask them for help out of "love" to stop some form of a disaster. Once the person consents to the Monks help, they form a link with the person who becomes their "lynchpin" through who they broadcast a revised version of the planet's history where the Monks have been there since the beginning. The psychic link gets passed through the bloodline of the Monks "lynchpin" and the link only ends if the "lynchpin" dies or the bloodline ends. (TV: The Lie of the Land)
At an unknown point in time, an incarnation of the Master encountered a planet ruled by the Monks. The Master ended the Monks hold on the planet by throwing their "lynchpin", a young girl, into a volcano. (TV: The Lie of the Land)
The Monks eventually started going after Earth as a potential new planet to rule. The simulated version of the Twelfth Doctor eventually discovered the truth and emailed a recording of the Monks simulation to the real Twelfth Doctor through the sonic sunglasses, warning him of the threat. (TV: Extremis)
After discovering the accidental creation of a world-ending bacteria, the Monks predicted a timeline where the Earth would be dead within a year and landed the Cathedral in a potential war zone in Turmezistan. The Monks showed a vision of the future they predicted to the gathered world leaders, including the Doctor as the President of Earth and offered to stop the event if the leaders consented out of "love". Each leader attempted to consent out of fear or strategy and were incinerated, but the Doctor managed to stop the bacteria on his own. Due to the Doctor's blindness, he would die in the explosion he set so Bill Potts consented in exchange for the Monks restoring the Doctor's sight. (TV: The Pyramid at the End of the World)
Over the next six months, the Monks quietly took over Earth, using Bill as their "lynchpin" and giant statues of themselves as transmitters to boost the signal of an alternate version of history where the Monks have always been on Earth and ruled the human race. The Monks locked the Doctor up in a prison ship, but he was eventually able to deprogram his guards and with the help of Nardole, recruit Bill to end the Monks rule of Earth.
After learning from Missy that wiping Bill's mind would be the most effective way to end the Monks rule, the Doctor led an assault on the Cathedral to find the Monks device that broadcasted the fake version of history and overwhelm it with the truth. After linking with the Monk controlling the device nearly killed the Doctor, Bill tried in an attempt to fry her brain so the Monks could no longer use her as Missy suggested. This failed at first as the Monk instead infected Bill's memories and used them to solidify the Monks control over the human race, but when Bill started broadcasting memories of the version of her mother she made up in her head, the Monk was unable to touch these pure memories. Ultimately, Bill's efforts broke the link and the Monks lost control of humanity.
With humanity turning on them, the Monks retreated from the Earth. Afterwards, humanity showed no memory of the Monks six month rule of them and the Doctor explained to Bill that the Monks erased themselves from human history. (TV: The Lie of the Land)