They appear in computer "RPG" games, and they've been a staple in tabletop games since the beginning of the hobby. I've discussed them a few times in old posts, and ways that they might be looked at. Alternate suggestions in other games have included a standardised range of "health levels", or penalty levels incurred for different degrees of damage.
But what I'm more interested in today are the alternatives for damage. In a social game you might look at political penalties that make it harder for you to accomplish social activities, or a pool of reputation points that absorb the insults and treachery of your opponents. In a digital "Tron"-style setting, you might have a pool of energy points that fulfill the dual purpose of absorbing incoming effects while being used to fuel attacks of your own (a precedent for this might be blood points in Vampire: the Masquerade/Requiem).
I'm trying to think of some other alternatives that might more closely link this absorption pool into a specific setting or genre of play.