Sometimes our heroes are faced with a foe they simply cannot withstand. Not only is it a fight they cannot win, it's a fight they have no hope of even surviving. They have two options: run away, or fight and die.
This enemy is usually rare and always notorious. Its very presence may inspire panic. Perhaps they're The Dreaded. Perhaps it's not an individual, but The Swarm or The Corruption. In any case, anyone who fights it is screwed, and anyone who could possibly end up fighting it knows it.
The key to this trope is that the threat is very powerful, but also impossible to predict. After all, if you can see it coming and avoid it, then it's not very scary, is it? But when it could show up almost anywhere, virtually without warning, and there's nothing you can do but hope to escape, then that's terrifying.
Of course, none of this will keep our heroes from fighting it sooner or later. If encountered early in the story, it may require a Heroic Sacrifice from one character to allow the rest to escape. If the sacrificial character is wounded, he may insist that I Will Only Slow You Down; either way, expect him to tell the enemy that You Shall Not Pass before making his Last Stand. If no sacrificial characters are available, then someone may show up unexpectedly and tell the heroes to Come with Me If You Want to Live. If the foe isn't fought until later in the story, it will probably be after the heroes have discovered its Achilles' Heel, or somehow dramatically increased their Power Level (through a Super Mode, Dangerous Forbidden Technique, or whatever), and it will still be a difficult fight. If it's fought both early and late, then the latter fight will be a Heroic Rematch.
In Video Games, may be a Hopeless Boss Fight where dying doesn't result in the plot continuing. If the only option you ever have is running, then it's probably an Endless Running Game.
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Anime and Manga
Pokémon: While Ash and co. are seasoned Trainers and well capable of defending themselves against wild Pokemon, there are two races that, when faced with, they'll run their asses off away from: Ursaring and Beedrill. It's notable that other Trainers usually share this philosophy as well. The one time Ash tried to fight back against a wild Ursaring, Pikachu landed one solid hit and the bear went ballistic.
During Bleach's Soul Society arc, Yoruichi advises the protagonists to run if they meet up with a Soul Society captain. Chad ignores the advice and ends up getting captured. Uryu uses a Dangerous Forbidden Technique to beat one, then loses immediately afterwards, in front of a second. Yoruichi, being a former captain, should have realized it'd be impossible for them to run from any captain that actually cared to follow.
During the Fake Karakura Town Arc, Gin Ichimaru informs Ichigo (who's been in a quite a bit of a slump up to that point) that he's in one of these situations.
Previously, when fighting Kenpachi Zaraki, Ichigo noted that running away is pointless because anybody strong enough to beat him is also going to be fast enough to chase him down.
This is the Karakura crew's entire strategy when Aizen busts into the real Karakura Town. With no spiritual powers whatsoever, Tatsuki, Keigo and others have no option but to run as Aizen slowly chases them down. Worst part? The only reason they survived was because Aizen was toying with them. If he had decided to kill them instantly, they would be dead.
One Piece had one of these come out of virtually nowhere during the Saboady Archipelago arc. It was probably the biggest Plot Twist ever seen since the manga started back in The Nineties.
Done even more significantly in the following arc, Impel Down. Luffy and the prisoners he's broken out are making their escape. Hot on their tails is the prison warden, Magellan. Normally taking on one guy wouldn't be a problem for Luffy and his allies, except Magellan's power is generating poison, and he's one of the most powerful characters seen at that point in the series, on his own domain, which he would protect at all costs.
Done to a less extent with Smoker. Before the timeskip his intangibility made him invulnerable to the Straw Hats and a very dangerous threat to them, so much that they just ran whenever they encountered him.
Serena Rinnen from Tower of God gives Phonsekal Laure this gem:
''Because I am not great or powerful like you, I spent 80% of my life just running away. Hundreds, no, thousands of times I have run, and because I have, I can be even more sure: Now is the time to run!"
An early Silver AgeCaptain America story has Rick Jones seeing Cap is in trouble and a Mook tries to silence him. Rick manages to take him down and a mate with a club as well, but as others are approaching, Rick remembers Cap's lesson that only a fool fights impossible odds. So, Rick decides he must make a break for it to get help and manages to escape. Just as the goons decide that they can subdue Captain America and get away before The Cavalry can arrive, Cap appears and takes them on considering for a One-Man Army like himself, two dozen to one is hardly impossible odds to him.
The Agents from The Matrix, providing the page quote. The sequels downgrade most of them to Elite Mooks at best for Neo after he's become The One, though they still remain significant threats to protagonists other than him and Smith remains the most dangerous of them all.
In the original Terminator, though the sequels tend to focus more on the Terminators fighting each other.
Phantoms from Final Fantasy: The Spirits Within. They're limitless in number and their touch is instantly lethal (via pulling out your soul); any fight against them is necessarily a delaying action until you can retreat... or a Last Stand.
Steel Inquisitors from Mistborn. Nigh-invincible Blood Knights who are definitely fighting at a higher Super Weight than the protagonists. The fight against one at the end of the first book is a Crowning Moment Of Awesome, and they remain the single biggest threat for the rest of the trilogy. One of Kelsier's Establishing Character Moments comes when he riles one deliberately in order to draw its attention away from Vin, though even he escapes as quickly as possible after that.
In some version of The Silmarillion, King Finwë was the only inhabitant of Fëanor's fortress of exile who did not run for it when a hostile Melkor approached to steal the Silmarils. He is slain for his trouble, and no surprise, since Melkor is the most powerful being ever created by God. Fighting him, for an Elf or a Man, is a losing proposition.
The Wheel of Time introduces several enemies like this in the first book alone. People don't enter the abandoned city of Shadow's Waiting unless they're forced to, because the evil that killed it is still trapped there and looking for new victims. People don't enter the Ways unless they're desperate, because the magic that created them was corrupted, and now they're haunted by the Black Wind That Steals Souls. And people don't enter the Blight unless they're suicidal, because, well...
The Labyrinth dragons in The Death Gate Cycle are so dangerous (Even by the standards of a Death World where the inhabitants face Everything Trying to Kill You - including the geography) that most people run for their lives if they even think that there's one in the area. If they do fight, it isn't to kill the dragon, it's to force the dragon to kill them quickly instead of slowly torturing them to death. The only Patryn in the history of the Labyrinth to fight a dragon and win is Xar.
All over the place in The Hunger Games. If you're a twelve year-old coming face to face with one of the careers, running is your only option. Heck, if you're Katniss coming face to face with the careers running and hiding in a tree is your only option. The trope is almost flat-out mentioned when Haymitch warns his tributes from trying to snatch supplies at the cornucopia, telling them they won't survive the initial bloodbath (though Peeta actually does in the book - in the movie he runs).
The tagline for the Goosebumps book "How to Kill a Monster" is "Step One: Run. Step Two: Run Faster."
In the Doctor Who two-parter "Silence in the Library"/"Forest of the Dead", this is all you can do against the Vashta Nerada. Seriously, the characters are never once able to take offensive action against them. The Doctor only gets them to back off by threatening to do something unpleasant based on his reputation, with no evidence in sight of how he would actually make good on it.
Your odds against the Weeping Angels from "Blink" and the "The Time of Angels"/"Flesh and Stone" two-parter, also Steven Moffat episodes, are only slightly better. By being very lucky, our heroes trick them into being stuck in the former and being destroyed in the latter.
In The Sarah Connor Chronicles, at the start of the second season, Cameron takes damage to her processor and reverts to her default orders to kill John Connor. Through the entire episode, the only thing the Connors can really do is run the hell away from her as she doggedly pursues them. The only thing that stops her is pinning her between two tractor trailers and cutting her central processor out of her head.
In Star Trek: The Next Generation, this initially looks like the only possible chance the Enterprise crew has to survive their first encounter with the Borg. It is subverted when the Borg are shown to be capable of running just a little faster and slowing the Enterprise down.
Cyrano de Bergerac: At act I, Actor Montfleury, who has dared defy Cyrano's prohibition to act, makes his choice after CadetCyrano claps a third time: He runs for his life. Later, The Bore also makes his choice, when after bothering Cyrano too much, he literally kicks his ass. It was the correct choice for both of them.
This, along with hiding, is standard procedure for any Survival Horror game that doesn't immediately start you off with some kind of weapon, at least until you're equipped to fight back. And even then, you will likely come across enemies that your weapons are useless against, making running your only option for survival.
Many Japanese RPGs that have random encounters love this trope; many dungeons are populated with sensible enemies for the most part, but as a player gets to the 2nd half of the game, they will often times find a Boss in Mook Clothing suddenly without warning. Many times, the game gives no indication of this until the hapless player attempts to fight it and dies a very horrible and frustrating death.
Final Fantasy I: Eye-type enemies, Dragon x4 fights in the last dungeon, WarMECH, to name a few.
Final Fantasy IV: Many of the monsters you find in the latter part of the last dungeon. The faces are some of the worst enemies in the game to fight; they give a lot less reward than most monsters near them and if you're low-leveled, they can easily kill you while spamming their spells. Even if you can survive battle against one, it is generally faster and easier to just run and fight something else instead.
Final Fantasy V: Grey Squirrels in a certain cave near the beginning of the game. They usually run away, but players are advised to do the same — they hit extremely hard (One-Hit Kill usually) should they decide to stick around and they are extremely hard to hit, and when you do hit them, you're only going to do 1 damage, although they only have around 5 HP. Unless at least one character is a Geomancer.
Final Fantasy VI: Doom Gaze while flying on the Airship in the 2nd world. He attacks randomly and if you're not prepared for him, you will very likely die.
Final Fantasy VII: If you didn't get the hints and/or talk to people later in the game, exploration underwater gets very dangerous if you run into Emerald WEAPON while unprepared. Similar for Ruby WEAPON, though the area you find him is rarely travelled so it is unlikely you'd come across him unless you meant to find him.
Bahamut from Final Fantasy III. You encounter him very early in the game, and if you don't run then he'll kill you on turn one.
The spider robot during the Dollet mission, which appears unexpectedly and chases the party all the way back to the evacuation point. It doesn't have a particularly powerful offense, but it's impossible to kill (except at one point) and it appears during a Timed Mission so you can't waste too much time fighting it.
The battle against Ba'gammnon in Final Fantasy XII in the Lhusu Mines. Some members of the party suggest fleeing from him and his gang, as beating them all in a head-on fight is extremely difficult without level grinding. If you exploit their weaknesses to certain status ailments (Sleep, namely), however, it's possible to win at a normal level.
In Thracia 776, as soon as Galzus steps out of the castle in chapter 6, it's time to haul ass in the other direction. Your units at this point have no business even starting to fight a swordfighter with Astra, Luna, and the three most relevant combat stats maxed out. (Besides, killing him with preciptious RNG abuse now prevents you from recruiting him at the end of the game.) The trick is in finding time to visit the houses in town for loot, and not letting your party get log-jammed too much trying to escape, before Galzus starts hunting you down.
Similar to Galzus, Vaida in Blazing Sword first appears as a ridiculously overpowered enemy thanks to her Spear being enchanted with Nergal's magic. It's highly recommended that you stay out of her attack range and focus on defense, since the mission only requires you to survive for 12 turns rather than defeat every enemy. And also like Galzus, while she is killable if you're clever and/or over-levelled enough, doing so prevents you from recruiting her later on.
Path of Radiance: Stepping past a certain threshold in chapter 11 prompts the Black Knight to mosey onto the field. Due to his stats and literal Plot Armor making him an walking wall of death, it'd be best to run to the escape point. Subverted the next time he pops up in chapter 24, where he will stand at the start of the map and just observe so long as no one goes into his range. Can possibly be played straight again when the time comes for Ike to duel with him - if his stats are simply not good enough to survive 5 turns with the Knight, you have the option of just booking it.
On a typical match of Team Fortress 2, if you see a Pyro tauting with the phlogistinator, in the middle of battle, and the weapon is glowing, run away. You can't kill the pyro unless you're a spy and backstabs him because the taunt gives him 75% of damage resistance during the taunt and when it's finished, the pyro will unleash crits. Which means if you're dumb enough to stay right next to him, you die. Painfully. Your body will become ashes.
You are doing good on the game and you turn around a corner and see... An ubercharged Heavy. What makes this worse is the visual effect that it gives to both the Medic and his partner, which is their entire body with it's team color with a very, very terrifying eye and expression.
Invoked in Halo when you first encounter The Flood. After the cutscene introducing them, the game changes your mission objectives to, quite simply, "Escape!". You can try to kill every Flood you see, but the cost in health and ammo will be high; you're better off running from most of them and fighting only when you can't avoid it.
Anyone who's played the Etrian Odyssey games long enough to encounter a FOE (which isn't very long) will tell you that unless you're massively overleveled for the floor you're on, you run the hell away from them. One particular case in Heroes of Lagaard is Salamox, who's nest you need to steal a MacGuffin from for a mission early on in the game. The chancellor, as well as the game itself, warn you that if you try to fight past it, you will die. They aren't kidding.
The Silent Hill series' infamous Pyramid Head is an example in all of his incarnations. He's completely invincible to any form of attack the characters have access to. Even when he is finally "defeated", it's less because anyone harmed him and more because his purpose was complete, so he committed suicide.
Silent Hill: Shattered Memories has no combat at all. You are told early on that running away from the enemy monsters is your only means of escaping the Otherworld.
"You have to run, Daddy. You can't fight them. Run!"
Silent Hill: Downpour channels the Otherworld Chases of Shattered Memories with the Void chases; When it appears you pull a 180 and run, throwing anything you can in it's path to slow it down.
In Prophecies, the last part of the "Great Northern Wall" mission has the character running away from a large group of Charr.
Later, one mission has you running from an army of Mursaat. If you attempt to fight, the party will die in seconds, due to the Mursaats' ridiculously powerful Spectral Agony attack. It is only after you have your armor "Infused" with spells to neutralize that attack that you can fight them on equal terms.
In Eye of the North, the quests in Kamadan, Kaineng, and Lion's Arch that take the character to the Eye of the North areas end with the player running away from destroyer groups.
Prince of Persia: Warrior Within has your encounters with the Dahaka except in the true ending, by which point you'll have a weapon that can kill it. All you can do is run away and try to find somewhere where it can't get to you.
In Metroid Fusion, the SA-X is ridiculously overpowered compared to Samus, being comparable to your power level at the end of Super and Samus being gimped by her new weakness to cold and a suit that doesn't protect her much. Your friendly AI commander actually gives you this instruction in as many words. As you progress through the game, you go from hiding, to running from, to preventing from following, and finally combat at the very end of the game.
True to the film series, the video game adaptation of Terminator: Salvation treats the T-600 series like this, especially the first few times you run into them. They're insanely durable, have mini-guns as their default weapons, and just keep coming. If you try and stay and fight the first time just one shows up, it will kill you. Later battles are only possible to win because you have lots of cover, explosives, and gun emplacements of your own.
Every enemy in Amnesia: The Dark Descent. You have no means of self-defense, so if you encounter a monster, your only hope is to run as fast as you can and hide until the monster stops chasing you.
The "Tycoon Wil" scenario in Saga Frontier 2 - it's a Hopeless Boss Fight in which all you can do is defend until the "Run" option comes up enough times for the ending cutscene to trigger.
The Hunter in Dead Space. You can attack it if you want, but it won't have any effect beyond briefly slowing him down. In order to escape, you have to lead it to the appropriate place to turn it into a Mutant Alien Zombie Popsicle; later on you have to do the whole thing again, but this time you get to Kill It with Fire.
Even before that, in both games, you're attacked by a swarm of Necromorphs without any weapons, stasis, or kinesis. Your only hope is to flee.
After the planetary shield drops in F.A.K.K.2, tough new enemies appear and Julie straight up declares that fighting them with her current weapons is futile. They technically can be killed, but it requires the better part of your current arsenal's ammo cap to down even one, so it's pretty awesome when your shiny new toys later in the game can grind them to hamburger.
In one of the third round of missions in Jedi Academy, a mutated rancor appears and you have to run away from it through the entire level (while fighting dozens of dark Jedi) until you can achieve an environmental kill. In an earlier level, you have to help prisoners escape from a rancor pit, usually by leading the rancor away from their group while they Run or Die.
The first two Paper Mario games have Clefts. If examined, you'll be warned that sometimes there's no shame in running, as they have so much defense that you can potentially find yourself incapable of harming them at all without the right items or moves ready. The truest example of this would be the "Invincible" Tubba Blubba from the first game, as if you battle him prior to finding out his weakness, you'll be completely incapable of harming him, thus making escape your only option.
There's one fight in the second game you can't win, though losing that one instead of running away doesn't result in Game Over anyway.
Mother 3. The Chimera Factory. The Ultimate Chimera. If it touches you or any member of your party, that's it. No Hopeless Boss Fight, just a cutscene where the Chimera chomps down and the screen quickly turns red before you get to the Game Over screen.
Fatal Frame final boss ghosts are like this whenever you encounter them prior to the end of the game. All of the games have a sequence where the main character must just run the hell away from the invincible ghost, lest she catch you and end your game instantly (interestingly, with the exception of the Kusabi, all of these ghosts are female). The third game has the Final Boss as a random encounter throughout (hint:RUN!), and also reintroduces the Kusabi from II in the Minakami Village areas. Though you can fight him off once or twice, it's really a better idea to just flee. He's freaking tough.
In the PS2 remake of Tales of Destiny, if you run around in one place with all your party members set to Auto for grinding purposes, bonus boss and Tales Of Destiny 2 antagonist Barbatos Goetia will eventually show up and exclaim that he's giving you the choice to run away or die. He's not kidding, since even if you do somehow manage to get his HP to zero, he'll just keep on fighting.
Tales of Phantasia has a Hell Lord monster infesting one area of the Slyph Mountain. If encountered, it will casts a powerful summon spell to kill off an entire party in one blow, forcing you to keep retreating until you solve a dungeon puzzle to get rid of it. Fortunately, running away from it is pretty easy to do.
World of Warcraft: In the dungeon Halls of Reflection, after beating a couple preliminary bosses the final "boss" consists of running from the Lich King until help arrives.
Pikmin 2 features the Water Wraith in one dungeon, which cannot be killed except with a Pikmin variety that cannot be brought in and can only be created at the last floor. Once it drops down, you haul ass to the exit.
In the Mega Man Battle Network series, randomly encountering a stronger version of a boss you've defeated in the past when you're not prepared for it is quite likely to become a case of this, as they tend to be much tougher then the previous version.
In a rather sadistic design choice, the option to flee was removed in the fourth game, making it entirely possible to suddenly end up stuck in a battle with something that can kill you in one or two hits on top of potentially having more HP then the final boss. They weren't nice enough to make you not get a game over for losing, either.
In Mass Effect 3, the Reapers, at least in the Galaxy Map. As you explore the Galaxy looking for Plot Coupons and War Assets, you run a risk of attracting the Reapers' attention every time you use your ship's sensors (without the sensors, you can't find anything). When the Reapers finally notice you, the Reaper Horn blares, and it's time to run. Getting caught by one of the Reapers is an automatic Non-Standard Game Over, no matter how good of an Ace PilotJoker is. There are only three missions in the game where you can directly engage the Reapers, and it's always with some kind of outside help from allied forces.
In a secret mission in StarCraft II: Wings of Liberty, you awaken a Protoss/Zerg hybrid just after setting the base's generators to overload. You don't have the firepower to take it down (and even if you did, it's invincible), so you're forced to run from it as you try to make your escape from the secret base. For the most part, it's an Advancing Boss of Doom, and it's possible to slow it down using an alien-tech "Chrono Rift Device", and other delaying actions are also possible. At one point, however, the invincible foe disappears from sensors, which is actually worse than knowing where it is; when it inevitably reappears, it does so very close to you.
In Persona 3, as you explore Tartarus, if you linger too long on any given floor, The Grim Reaper will show up. He can be fought, but it's a very, very bad idea.
At the end of the prologue in Baldur's Gate, the beginning-level Player Character and their mid-level mentor Gorion are ambushed by the Big Bad Sarevok and some minions. Gorion stays to fight while telling the Player Character to flee, which they do. He kills or disables the others and holds off Sarevok for a while, but hasn't really got a chance. When you actually fight Sarevok at the end of the game, at least he's got only about two to two and a half times your levels...
At a couple points in Condemned 2: Bloodshot, the interface itself says "RUN FOR YOUR LIFE!", most memorably when you first encounter the rabid grizzly bear.
Nemesis and Tyrant T-101 come close to this in Resident Evil, but you actually can fight them. The Ustanak in the sixth installment plays this very straight in his first appearance: When he shows up, you run your ass off.
In Clock Tower when you hear this, you run for dear life until you find either a place to hide or a means to drive him away.
Double Subversion in Another Gaming Comic, during a Matrix-pastiche arc. The Agents are hyped up as unkillable, unstoppable monsters who, if you do somehow manage to kill them once, will just come back again. Since the protagonists are highly talented power gamers, their first encounter with an Agent results in the agent getting one (high-damage) hit in and then getting vaporized. The heroes then proceed to run away very quickly, explaining it to Nuclear Dan as they go.
Contessa, of Worm, is notable in that she operates extremely quietly, acting as a censor of information about Cauldron, up until the point when Cauldron is exposed to the superhero community, at which point she begins to eliminate leaks more openly, notably incapacitating Faultline's crew, an experienced mercenary team that has previously taken on Myrrdin and Chevalier, in less than twenty seconds. Protectorate policy on facing her is "Just run," and when Weaver ignores this directive, Contessa hands her a total defeat.
Kagerou Project: The song Headphone Actor is about a teenage girl finding out via the radio that the world is about to end, and a voice coming from her headphones - her own voice - tells her to do this. It's futile.
Its sequel Yuukei Yesterday (and added context from the manga) reveal that the song is actually a metaphor for Takane Enomoto sprinting to her (apparently terminally ill) crush's bedside, because she fears it'll be too late. Either way, she ends up dying.
The Creepy Pasta "Never Stop Running" is about a video game programmed so that if your character stops, for whatever reason, they are smashed to paste by an Advancing Wall of Doom.
A similar one has this happen with a pack of wild dogs, and Every time the character dies, they become slower and slower, until they are immediately eaten when the game starts.