"Alright, alright. You're makin' a nice little dent in my undead army. That's the beauty of the undead — if you kill them they're just dead again, and I just fire up the mojo, and make them re-undead and they're back to gnawin' at your hind fat."
The idea of a creature which gets back up after it appears to be dead is quite popular in fiction (cases in point; Not Quite Dead monsters and protagonists). It makes them seem more intimidating, ensures that the hero is challenged, and in some cases keeps things more family friendly by making sure that no one actually dies.
In Video Games, a Reviving Enemy is what happens when an enemy who is supposedly defeated is able to get back up after awhile and start fighting the player again. Defeating these enemies for real may be as simple as simply beating them a second time (expect them to Turn Red and/or go One-Winged Angel after their first knockout), or it may require use of special tactics (such as attacking them when they appear down) or using a specific weakness against them. The term "defeat" can also be taken loosely; enemies which are capable of escaping normally-permanent restraints or traps can also count.
Note that these shouldn't be confused with Respawning Enemies, which can be defeated normally but are constantly being replaced by their fellows.
These tend to be useful for Survival Horror games, where enemies work best when they're harder to fight off permanently (and having them coming back to life is always good for a scare), or adventure games (particularly Metroidvanias) where it justifies old areas being inhabited without needing to resort to Respawning Enemies when the player returns and obtaining the means to finish them off can be a Beef Gate of sorts. Defeating them might take the form of a Stock Video Game Puzzle where events in the plot get rid of them rather than giving the player the means to do so themselves.
This might involve I Surrender, Suckers for human(oid) enemies and Deader than Dead for supernatural ones. It often overlaps with Roaming Enemy (particularly in adventure games) so that the player can avoid them. Robotic and Undead enemies are especially prone to this. Sister trope to Attack Its Weak Point and subtrope of For Massive Damage. Contrast Respawning Enemies. If they are the sort with a specific weakness, means for defeating this sort of enemy can include an Achilles' Heel (which may be a Weaksauce Weakness) or Finishing Move (particularly a Coup de Grāce).
Enemies which have a particular weakness but are simply invincible if it isn't exploited fall under Achilles' Heel rather than this trope. Remember that the important part of this trope is that they can get back up after being taken down, not the details of how or how to prevent it.
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Stalfos (or at least the most powerful types) in The Legend of Zelda often need to be finished off by blowing up their bodies with bombs to stop them from reanimating (this is usually only needed to clear rooms that don't open doors or spawn chests until all the enemies within are defeated, since the games make use of Respawning Enemies).
In The Legend of Zelda: Majora's Mask, the two Stalfos knights which the King of Ikana sends out to fight you have to be defeated in ordinary sword combat first...and will get back up again if you don't quickly finish them off by reflecting light onto them with your Mirror Shield. The same applies to the King himself.
In The Legend of Zelda: Twilight Princess, the Twilight monsters Link regularly fights do not revive by themselves, but if Link is facing more than one, the last monster will emit a paralyzing shriek that revives its fallen comrades if Link cannot defeat them simultaneously.
Solomon Grundy does this in Batman: Arkham City. After damaging the generators powering him a few times, he'll fall to the ground and you'll be given the opportunity to "Finish Off Grundy." Batman will jump on top of him, only to get grabbed by Grundy and have to break free of him through a Quick Time Event.
Vampires in Soul Reaver - Nosgoth vampires go limp when defeated but have to be submerged in water, burned or Impaled with Extreme Prejudice or their regeneration will kick in a moment later. One of the bosses is actually impaled by someone else and will rise to fight you if you pull out the spear.
A common feature of enemies in the Call of Duty series. Sometimes after you kill an enemy, instead of dying normally, the enemy might simply fall down into a wounded state. At first glance, it appears the enemy's dead, but they will pull out a pistol and try and drag themselves to relative safety. They'll die eventually, but it can be a problem for a player who's in a hurry or not very attentive, since rushing forward can end with them getting shot in the back by the Not Quite Dead mook.
Multiplayer often has a perk/deathstreak called "Final Stand". When equipped, instead of dying when their health reaches zero, the player will sometimes fall into a "last stand" state where they are injured and reduced to a One-Hit-Point Wonder but able to defend themselves with a pistol and can regenerate enough health to get back up if they survive.
AxeZombies in Blood are of the Not Quite Dead variety. Any blow past a certain damage threshold that isn't fatal will just briefly knock them down.
Zombies in Quake will get up again; you need to gib them with explosives to separate their bodies.
In various Super Mario Bros. games, the undead Dry Bones easily collapse underneath your Goomba Stomp, but will get back up after a short while; you generally need something stronger (like a turtle shell) to defeat them for real.
This carries over into Paper Mario, where Dry Bones will automatically revive after 1 or 2 turns of being down unless you Kill It with Fire or end the battle before then.
Globins in The Binding of Isaac collapse into inanimate piles of meat when killed; however, they'll regenerate after a few seconds unless you destroy the meat pile as well. Stronger Globins can actually flee while in this form, making them tougher to bring down permanently.
In Mega Man X3, the mini-bosses Bit and Byte and Vile can be defeated with any weapon, but will survive and reappear as main bosses in the final stage. However, if they are killed using the specific weapons they are vulnerable to, they will be Killed Off for Real and different bosses will be appear in the final stage instead.
Mega Man 4 has the skeleton Mooks in Skull Man's stage. Attacking them with most weapons will cause them to collapse, but get up again (although it will do damage). However, a charged shot will put them out in one hit.
In Wario World the larger enemies must be punched two or three times into unconsciousness, at which point they can be thrown around or used for attacks. However, if left alone they will get back up and require the same amount of hits to knock down.
In Prince of Persia 2: The Shadow and the Flame, skeletons will revive with full health a few seconds after being slashed into a pile of bones. The same applies to the brown skeletons in the SNES version of Prince of Persia.
All of the enemies in Prince of Persia: The Sands of Time will revive after being knocked down unless you finish them off with the dagger. Later on you lose the dagger but get a sword that one hit kills them.
One of the enemies in the "Harbour" stage in Contra 4 does this. It starts out as a humanoid robot, destroy it once and it collapses onto the floor, only to get back up as a walking cannon or a dog mech. Destroying it a second time puts it out for good.
Tomb Raider Underworld features Norse zombies called Thralls that you need to body-slam to shatter, or they'll pick themselves up off the floor and continue trying to kill you.
City of Heroes: The higher-end members of the Freakshow do this with annoying regularity. And in the final mission of the Imperious Task Force, Nictus-infused Romulus will do this three times before he stays down for good.
Several bosses in World of Warcraft will revive one another if you don't kill them within a handful of seconds of each other.
In Dwarf Fortress, if you're brave and/or stupid enough to build your fortress in an evil biome, the corpse of anything killed will, after a short period, rise as a zombie. "Anything killed" includes zombies, of course, meaning the only way to permanently kill anything is to encase in in ice or obsidian, or dissolve it in magma. Taking this trope to ridiculous extremes, even severed body parts will come back to life, meaning one dead dwarf can quickly become a one-armed zombie and a zombie arm, then a one-armed headless zombie, a zombie head, a zombie arm, and several zombie fingers.
Role Playing Game
In The Elder Scrolls V: Skyrim some enemies will surrender and might flee or invoke this trope (there's no real penalty for finishing them off). Trolls and Frost Trolls' animation for this has them play dead (referencing the fact that they have fast regeneration) which could invoke this trope if the player doesn't notice that they've got some health left.
Spriggins also count, since they can restore themselves to full health.
Any enemy can become this when there's a character who can raise the dead around, since unless you take steps to vaporise the corpse (e.g. kill them with lightning when you have the appropriate perk or raise them yourself) you'll have to kill them again (anything being raised this way collapses into ash when they or their "master" is killed). This makes necromancers and vampires rather annoying.
In Mass Effect 1, Krogan enemies would always fall down dead after their HP hits zero, but would sometimes randomly revive get back up with a little bit of extra health.
The Reanimated Horde for the Diablo II expansion pack had a chance of rising again after you'd killed them (although this could only happen a finite number of times and wouldn't always happen). The only way to be sure was to Kill It with Ice, which would cause the body to shatter and evaporate.
In Dark Cloud, all of the skeleton-type enemies (Master Jacket, Corsea, ect.) have a chance of getting right back up after falling over dead. It can happen multiple times with a single foe.
Regenerating monsters in Dungeons & Dragons need to be killed by a form of damage that they can't regenerate away, or they'll come back to life. For example, trolls can only be permanently killed by using acid or fire (or by temporarily killing them in some other way, and then inflicting acid or fire damage on the remains).
The Ghost People of the Sierra Madre from the Dead Money DLC of Fallout: New Vegas will revive after several seconds unless killed by dismemberment.
Troll-class enemies in Nethack will revive from their corpses (including Olog-Hai), so they need to be dealt with in such a way that they either don't produce a corpse (drowning works; throw the corpse in a lake, watch it revive and drown immediately), the corpse is contained enough to rot before it revives (store it in a chest), or you eat the corpse quickly enough (and not get "The bite-covered troll rises from the dead!").
Skeletons within the Catacombs in Dark Souls will reassemble themselves ad infinitum when you defeat them. There are two ways to defeat them: One, defeat them with a Divine-upgraded weapon, or a unique weapon that deals Divine damage, or find the local necromancer and defeat him. Post 1.04 patch, where skeletons finally give souls, whether or not you receive souls for your kill is the only indication that they're dead for good.
There is actually a third way to kill them permanently. The stage is arranged such that there are many steep falls throughout. Simply kicking them all the way to fall off a cliff will actually kill them, since lethal fall kills also applies to them.
Epic Battle Fantasy 4 has an enemy called the Spirit, which enters battle with Auto Revive. If the player fails to use a Status Buff Dispel it before defeating it (or waiting enough turns for the auto-revive to run out), it comes back to life again.
In the Rogue-Like Dungeon Crawl, the unique lich Boris never dies permanently. Whenever you kill him he'll respawn somewhere else in the dungeon.
Undead in Final Fantasy Tactics and Final Fantasy Tactics Advance do this unless you use Revive, a special skill, or Phoenix Downs on them (if they're already temporarily killed it's a guaranteed removal and for the latter two a chance of a One-Hit Kill if they're not). If you don't, you'll only win a "defeat all enemies" goal if every other monster is dead and all undead are waiting to revive at the same time.
In the REmake of Resident Evil for the Gamecube, zombies must be killed via decapitation or have their bodies burnt. Otherwise they'll come back to life as the much more powerful Crimson Heads after a set amount of time.
Skeletons in Clive Barker's Undying are only temporarily stunned by normal weapons. After a while, they get up again. Only using Invoke on them, or slicing them with the scythe several times can permanently kill them.
Skeletons would also constantly recover in Nocturne. Only by cheating and using a flame-based weapon can you permanently destroy them.
The shibito in Siren cannot be killed, due to the red water permeating the village where the game takes place. They can only be incapacitated just long enough for the player to make their escape.