I have heard of games (e.g. Dungeon World or Monsterhearts) being called "Powered by the Apocalypse", or "PbtA".
What does this mean?
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In common parlance, a 'Powered by the Apocalypse' game is a game that uses the same resolution engine of Apocalypse World, by Vincent D. Baker, or is somehow recognizably inspired by or similar to it.
In other word, it is a way for people to categorize Apocalypse World hacks.
Despite this being the perceived definition, commonly used in a large part of the Internet including the tag description in this site and Wikipedia, PbtA is in reality something different.
"Powered by the Apocalypse" isn't the name of a category of games, a set of games' features, or the thrust of any games' design. It's the name of Meg's and my policy concerning others' use of our intellectual property and creative work.
Here's the policy:
If you've created a game inspired by Apocalypse World, and would like to publish it, please do. If you're using our words, you need our permission, per copyright law. If you aren't using our words, you don't need our permission, although of course we'd love to hear from you. Instead, we consider it appropriate and sufficient for you to mention Apocalypse World in your thanks, notes, or credits section.
It's completely up to you whether you call your game "Powered by the Apocalypse." If you'd like to use our PbtA logo in your game's book design or trade dress, ask us, and we'll grant permission for you to do so. This isn't a requirement of any sort.
Again, "Powered by the Apocalypse" isn't the name of a kind of game, set of game elements, or even the core design thrust of a coherent movement. (Ha! This last, the least so.) Its use in a game's trade dress signifies ONLY that the game was inspired by Apocalypse World in a way that the designer considers significant, and that it follows our policy wrt others' use of our creative work.