Fourth wall
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The fourth wall is a theatre term referring to the audience. This originates in the idea that there are three walls on a stage: one on the back, one to the left, and one to the right, as well as an imaginary fourth wall in front that contains the players within their play. To "break the fourth wall" means to show awareness of the audience or other things outside of it.
The fourth wall was famously broken in episode 7 of The Daleks' Master Plan, "The Feast of Steven", in which the First Doctor wishes the viewers a "happy Christmas": "Oh and incidentally, a happy Christmas to all of you at home." This is the only case in the series proper in which a character explicitly displays knowledge of being on TV; all other cases of fourth wall breaking involve the characters seeming to talk to or perform for the camera, but nothing that cannot be explained by another character or a mirror being in the position the camera is occupying or a character talking to them self, and nothing which displays knowledge of being a TV character.
At the end of The Invasion of Time, the Fourth Doctor breaks the fourth wall by grinning mischievously to the camera.
In the final scene of The Caves of Androzani, the Sixth Doctor looks directly at the camera and says "Change my dear; and it seems not a moment too soon".
At the end of The Trial of a Time Lord's last story, The Ultimate Foe, the Valeyard breaks the fourth wall by looking directly into the camera and laughing.
In Remembrance of the Daleks part 2, as Ace walks out of the B&B, the TV announces, "A new sci-fi series, called Do..." before cutting to the next scene.
Just before her death in Asylum of the Daleks, Oswin Oswald says, "Run you clever boy. And remember," and turns to the camera.
At the end of The Snowmen, the Eleventh Doctor breaks the fourth wall by saying "Watch me run" to the camera.
In the proms special, Music of the Spheres, the Tenth Doctor breaks the fourth wall by addressing the audience at the Royal Albert Hall.
At the end of the Series 1 finale of K9, The Eclipse of the Korven, K9 breaks the fourth wall by looking directly at the camera and saying, "Affirmative".
In Deep Breath, after the Half-Face Man has fallen from his "escape pod" and been skewered on a spike on the top of the Elizabeth Tower, the Twelfth Doctor looks directly into the camera as the question is posed: "Did the robot self-destruct or is the Doctor a murderer?"