The Vault (The Pilot)

The Vault, hidden beneath the campus of St Luke's University in Bristol, was a Quantum Fold Chamber which contained. The Twelfth Doctor made a oath to protect the chamber for a thousand years.

Like some other examples of Time Lord technology, the chamber was bigger on the inside. The interior resembled a nondescript room with some tables and chairs and, even though it was a sealed unit located underground, windows that allowed light to enter the chamber. A centrepiece of the chamber was a piano with which Missy amused herself.

The chamber had Gallifreyan markings all around it and on both sides of the doors.

History
A long time ago on a planet, the Doctor was given the duty of executing, by use of a technology able to kill a Time Lord off completely. However, the Doctor fiddled with the machine and it didn't kill her. The Doctor had sworn to watch over her body for 1000 years, but he did not say that she would be dead. (TV: Extremis)

By the 2010s, the Doctor and Nardole had spent at least seventy years guarding a vault on Earth (TV: The Pilot) that became the prison for Missy, while also working at St Luke's University, the premises beneath which the vault was located. (TV: The Pilot, Extremis)

When the Doctor and Bill Potts encountered the puddle of sentient oil in the Doctor's study, they and Nardole escaped in the TARDIS and landed under the university to check on the vault. The team fled the campus in the TARDIS after they learned that the oil was not interested in the vault. (TV: The Pilot)

After being told by the Doctor to "leave [him] alone", a frustrated Nardole voiced his frustrations to himself while standing guard at the chamber before being interrupted as Missy started heavily knocking the door. Though acknowledging that the Doctor had been "distracted" by Bill, Nardole asserted to Missy that he would still stand guard and not let her escape. (TV: Thin Ice)

After saving Bill and her friends from a creaking house filled with Dryads owned by a mysterious landlord, the Doctor returned to the chamber with food to take his shift. He found Nardole checking the security of the chamber doors. Missy began to play Für Elise on a piano which was gifted by the Doctor, against the wishes of Nardole. Nardole was dismissed by the Doctor, who went on to tell Missy about his day. Hearing about people being eaten, the tune changed to Pop Goes The Weasel and the Doctor entered the chamber. (TV: Knock Knock) Despite the chamber being effectively a prison, the Doctor was able to enter the chamber to visit Missy at will. (TV: Knock Knock, The Lie of the Land)

While outside the vault, the Doctor received an email on his sonic sunglasses from a virtual Doctor and, in a one-sided conversation, asked Missy for her help. He also admitted that he was blind. (TV: Extremis)

After the Monks succeeded in invading Earth, the Doctor and Bill consulted Missy inside the vault about how to defeat them where she was playing Gnossienne No. 1 on the piano. At first, she played a game of Hot or Cold when he attempted to find the answer himself. Then she played The Entertainer on the piano and then told him she had dealings with them before and that he would have to kill the lynchpin which was Bill Potts.

She and the Doctor later sat in the vault and conversed, during which she was thoughtful about her past killings and murders, which she realised she actually knew the names of, the Doctor realises she might be turning good. (TV: The Lie of the Land)

When Nardole was sent back to the TARDIS during a trip to Mars to fetch something for the Doctor, she took off by herself, with Nardole onboard, because she sensed danger and refused to return, Nardole had no choice but to consult Missy and bring her out of the vault to steer the TARDIS.

When the Doctor found out, he told her he would have to put her back in the vault again. (TV: Empress of Mars) However, instead of going through with this, he chose to keep her in the TARDIS, to perform maintenance on the ship's systems, after bio-locking her out of the controls and making sure she could not exit the doors on her own. (TV: The Eaters of Light)