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The d20 system is a Game System for Tabletop RPGs created by Wizards of the Coast and premiering as the system used for the third edition of Dungeons and Dragons. What makes the d20 system unique is that it was released under the Open Game License (OGL), which was created by Wizards specifically for that purpose. Under the terms of the OGL, developers are granted permission to copy, use, modify, and redistribute the system -- effectively making it analogous to open source software. This means that anyone can create their own RPG using the d20 system, and even sell it for profit, without running afoul of copyright issues.
The core mechanic of the d20 system is rolling a twenty-sided die (the eponymous d20), adding a "modifier" to the result, and comparing that to a "difficulty class" (DC) to determine whether the action was successful or not. The d20 provides randomness, the modifier represents the character's skill, and the DC represents the difficulty of the action being attempted. If the total result (d20 + modifier) is greater than the DC, then the action succeeds. Modifiers are dependent on the character's stats, and both modifier and DC may be altered by situational bonuses or penalties (for example, if your target is lying prone, ranged attacks become harder, but melee attacks become easier). The most important modifiers are generally The Six Stats, at least one of which will be applied to almost every roll, but other check-specific stats will also be included. Also of note is the "opposed check" -- when two characters directly oppose each other (one character tries to sneak past another, for example), then both will make the relevant skill check (eg, stealth vs. notice), and the higher result wins.
Outside of this core mechanic, the system is very flexible, and include everything from D&D's classic Class and Level System to Mutant & Mastermind's entirely Point Buy character creation. This flexibility is a large part of the reason for the system's popularity -- it allows different systems to cater to different tastes, without really changing how the game is played, as "d20 + modifiers vs. DC" remains constant.
- Dungeons and Dragons. It was created for D&D third edition, and kept on for 3.5, but 4e used a different (non-open) license. Numerous third-party systems are intended to be additions to (or at least compatible with) 3e D&D, rather than outright replacements.
- Iron Heroes, a system from Fiery Dragon Productions, is dedicated to more tactical fighting than vanilla 3e D&D. Borrows heavily from action hero and fantasy movie tropes.
- Arcana Unearthed/Arcana Evolved by Malhavoc Press is a major rewrite with new spells, new classes, and new types of magic. Generally highly praised.
- Pathfinder is a "D&D 3.75 edition" by Paizo Publishing, notable for being though of as made specifically in response to 4e, when in reality it was in production before 4e was even announced. Written to be compatible with all of 3.5's material, it also attempts to fix some of the remaining flaws in the system, though the quality of its changes is a contentious subject among fans. Successor to the 3.5 throne, it is highly successful commercially and widely supported by third parties.
- The Legend Game System actually began as a series of house rules for 3.5 before the writers decided they had enough material for a full game.
- Mutants and Masterminds, a superhero-themed Point Buy RPG, uses the d20 system for its core mechanics.
- Northern Crown, a North American fantasy setting by Atlas Games.
- D20 Modern and d20 Future, both developed by Wizards of the Coast, were attempts to bring the d20 system out of the Heroic Fantasy setting iconic to D&D proper.
- True20 is a simplified iteration of 3.5 from Green Ronin Publishing. Adapted to a wider variety of genres than D&D, it's a Spiritual Successor of sorts to d20Modern. Its major selling point is ease of use and speed of play.
- D20 Rebirth is a fan-made system in development, meant to serve as an alternative to Pathfinder while retaining the same "3.75 Edition" feel.
- Microlite 20: A free, extremely streamlined and rules-lite version of the d20 system, designed to be compatible with existing d20 monsters and adventure modules.
- A large number of unrelated systems and/or settings have d20 variants: