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Inspired by this question (and a comment underneath it), I thought we should have a solid answer on the site for this question:

Can a character at 0 HP take damage?

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Yes. This is implied by the rules on death saving throws (PHB p.197), which state in part (emphasis added):

If you take any damage while you have 0 hit points, you suffer a death saving throw failure.

Note that damage while you are at 0 hit points doesn't reduce your hit points, though:

A creature's current hit points (usually just called hit points) can be any number from the creature's hit point maximum down to 0. (PHB p.196)

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Yes, but damage is a different concept at 0 HP than at nonzero HP.

Hit point mechanics change when a character reaches 0 HP. The character can still be hit with attacks and other damaging effects, but they don't take additional points of HP damage. Instead there are two variations:

1. Failed death saves

Instead of taking damage, a character at 0 HP that is hit with an attack immediately fails a death save (two death saves on a critical hit). This brings a character closer to death, but does not change their HP from 0. Notably, these effects are similar to the results of actual death save rolls: less than 10 causes a single failure, and a 1 causes two failures.

People would probably not refer to a failed roll as damage in quite the same sense that they use damage when referring to losing HP, but this is the effect attacks which would cause damage to a >0 HP opponent have on an opponent at 0 HP.

2. Massive damage (specifically for a target at 0 HP)

Massive damage also does not deal individual points' worth of damage but instead simply causes death, with no intervening steps. This mechanic is mathematically identical to massive damage at >0 HP, but is marked out as a specific case in the PHB.

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    \$\begingroup\$ Massive damage is not the same as three failed death saves. It kills you outright. If it were the same as three failed death saves it would interact with some types of game mechanics connected to death saving throws and saving throws in general. Massive damage resulting in death does not intreact with these mechanics \$\endgroup\$ – David Coffron Sep 30 at 19:50
  • \$\begingroup\$ @DavidCoffron I'm referring specifically to the case of massive damage to a target at 0 hp (distinct in PHB from other massive damage scenarios), and was describing it by analogy (hence my use of the word effective). It seems to me to be mechanically equivalent to 3 failed death saves by any vector, after which the character is completely dead. Other mechanics for taking damage while at 0 hp all involve death saves, is your critique about the other case of massive damage? Or am I still misunderstanding? \$\endgroup\$ – Upper_Case-Stop Harming Monica Sep 30 at 20:00
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    \$\begingroup\$ Didn't downvote, but also don't like likening Massive Damage to an instant 3 failed death saves. It's not immediately effective you're using an analogy and on a simple question like this it only serves to muddy the waters on what actually happens ("Wait. Is it actually 3 failed throws or just similar?") In addition to the earlier commentary on the fact that there are mechanics that impact how death saves work, and they have no business around a massive damage situation \$\endgroup\$ – DeadChex Sep 30 at 20:05
  • \$\begingroup\$ @DeadChex TBH, I don't see a meaningful difference since I was not referring to death save mechanics except to reference the "all the way dead" state, which is the only thing 3 failed death saves does, but it's not phrasing worth keeping if it adds confusion or upsets people. \$\endgroup\$ – Upper_Case-Stop Harming Monica Sep 30 at 20:15
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    \$\begingroup\$ @DavidCoffron: can you give an example of an effect that would make a difference? Could a Legandary Resistance let you choose to not have failed one of the death saves, if massive damage was defined as mechanically equivalent to 3 failed DSes? Or are you thinking of some specific spells or class features that do things with death saves specifically? (I don't remember the specific details of any, but it's not like Upper_Case is saying it would count as having made and failed them, just as if you had failed them.) Still not a fan of the wording because it raises the questions, though. \$\endgroup\$ – Peter Cordes Oct 1 at 5:53

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