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A type of Temporal Paradox. The name comes from the most famous variation - namely "what would happen if you travelled back in time and killed your own grandfather?", but also applies to anything that happens while time travelling that should logically make the original time travel impossible or unnecessary.
If you kill your grandfather or destroy the time machine in the past, you should never have been born or never been able to travel into the past in the first place. If you kill the evil overlord while he's a child, you shouldn't have any reason to travel into the past to kill him, since in the changed present he was killed before he destroyed your village. So killing your grandfather causes you to not exist to kill him. Which means he survives. Which means you do exist and proceed to kill him. And so on.
This type of paradox can be ignored if the type of time travel being used involves Alternate Universes instead.
If the universe runs on Stable Time Loop, this type of paradox is impossible. Since all changes that will have been going to happen "already" happened, you can't cause a change that will negate itself.
May lead to My Own Grampa, though in this case the person killed wasn't originally your grandfather anyway.
Examples:
- The plot of the first Back To The Future may be the most well known example, even though it's A. not Marty's grandfather it involves and B. he doesn't kill him, but rather accidentally takes his place as his mother's object of affection. The rest of the movie has Marty trying to correct things before he's erased from existence.
- In Metal Gear Solid 3, if you kill one of the bosses instead of just knocking him out, you get the "TIME PARADOX" Nonstandard Game Over. The reason is quite meta: you killed a boss which starred in a previous game, because 3 is a prequel to 1 and 2, which means you killed a boss in 3 which exists in 1 and 2, so with that boss dead he shouldn't exist in 1 and 2 anymore, so...
- In an episode of Futurama, Fry goes back to Roswell in 1947 and accidentally gets his grandfather killed in an atomic blast while trying to avert this trope. He doesn't stop existing because he also ends up doing his grandmother, becoming his own grandfather. Or, as he put it, "I did the nasty in the pasty."
- Narrowly averted in The Legendof Zelda: Oracle of Ages; Ralph attempts to destroy his own ancestor Queen Ambi (who is possessed), knowing that it will remove him from existence. Subverted when she ends up kicking his ass instead.
- In Heinlein's Time Enough for Love, Lazarus Long just glosses over the possibility of creating a paradox while time-traveling by saying that it's impossible to create one. So he has sex with his mother, meets his younger self, enlists in the Army and fights in World War I.
- In Door into Summer, this is boiled down to the time-traveler protagonist waiting just outside of a room where he also is prior to his time-traveling activities, and briefly wondering what would happen if he ran in and slashed his counterpart's throat. He doesn't do it, of course, because that would be stupid and accomplish nothing, but he notes in present tense that he still hasn't figured it out.
- Inverted in Stargate Continuum, where Cam Mitchell winds up going back in time, and eventually (ten years down the road) keeping his Grandfather alive as a way of setting right what Ba'al had messed up.
- In one of the demo videos
 for the RTS game, Achron, they build a mech, then send it back in time to destroy the factory that built it.
- In Chrono Trigger, Marle disappears from existence due to her being mistaken for her ancestor Queen Leene who was kidnapped at the time she landed in the past, and since they stopped looking for Leene thinking they had found her, she was killed, thus causing Marle to not exist. Fortunately Leene hasn't been killed yet, so our heroes are able to go rescue her, which causes Marle to start existing again. While this is clearly a Grandmother paradox, everyone seems to remember that she existed at least long enough for her to cause herself to have never existed.
- For the Time Lords, Grandfather Paradox is an actual person who went back and, yes, killed his grandfather, which doomed him to a sort of undead temporal limbo. He's the Time Lord equivalent of the Bogeyman, and the splinter group/terrorist cult Faction Paradox considers him their spiritual leader, partly cause it pisses off the Time Lords.
- We actually meet him. He's kind of a cosmic embodiment of Future Me Scares Me - he's everybody's evil future self.
- In the newer series, The Master constructs a Paradox Machine specifically to evade the consequences of this paradox, as his army of psychopathic laser balls happen to be the descendants of Earth.
- In Johnny And The Bomb, Bigmac suggests going back in time to kill Hitler. Johnny warns him of the dangers should he accidentally kill his grandfather, but Bigmac says it's safe since his grandfather doesn't look anything like Hitler. (Fortunately, by the time they obtain actual time travel, he's forgotten the plan.)
- A Newgrounds cartoon "Grandbunny Paradox" made fun of this. It featured a bunny and a stick figure. The bunny went back in time to kill his grandmother and finds himself turned into a sheep, because his grandfather married a sheep instead of a bunny. The stick figure decides to do the same and kills his grandmother only to find himself turned into a tomato. He doesn't like being a tomato so he goes back and shoots the guy who sold him the gun to kill his grandmother...only to find himself now holding grenades.
- Oh so inverted by the Lost episode "The Variable".
- Genius The Transgression's stance on the subject: "And yes, if you kill your own grandmother before your father is born, you will cease to exist. The universe, it turns out, doesn't care that much if your grandmother gets shot in the head and there's no shooter. You still go poof."
- The Charles Stross novella Palimpsest has a twist: killing your own grandfather is the initiation rite for the Time Police.
- Paradoxically inverted in "Grandpa", a short story by Edward M. Lerner. In it the protagonist, Professor Fitch, survives two assassination attempts by his grandson, and preempts a third by deciding not to have children.
- In The Adventures Of Sonic The Hedgehog, one episode in the four-episode arc about the Chaos Emeralds starts with Sonic foiling a plot by Robotnik to prevent his ancestors from marrying and thus eliminate Sonic from the timeline. Sonic succeeds in sending Robotnik packing, but then causes the paradox himself by ordering a chili dog from his maternal ancestor, causing his paternal ancestor to become impatient waiting to be served and leave. After Sonic disappears, Tails solves the paradox in about a minute by forcing the meeting to happen.
- Averted (and explained too!) in Terry Pratchett's 'The Lost Continent'. Because, in fact, history already depends on your treading on any ants that you happen to step on.
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