these are some i know of:

---

phpBB

phpBB Logo.

phpBB is by far the most popular and better forums you can have in open source..

phpBB is a high powered, fully scalable, and highly customizable Open Source bulletin board package. phpBB has a user-friendly interface, simple and straightforward administration panel, and helpful FAQ. Based on the powerful PHP server language and your choice of MySQL, MS-SQL, PostgreSQL or Access/ODBC database servers, phpBB is the ideal free community solution for all web sites.

phpBB version 2 focuses on a professional-quality modular design, high security, multiple-language interface, support for a multitude of databases servers and complete layout customisation, all with a low execution overhead.

The security of phpBB has been disputed recently. Some people say that phpBB is an insecure product, while others say that it doesn't have more security problems than other forum systems. This can be argued, though, but the team has almost always responded fast and properly, trying to avoid future problems for board administrators.

phpBB is a widely popular Open Source forum that works well, have a simple user interface, can be customized and support many languages. Just the possibility to customize phpBB with skins/themes has made it very popular. There is a lot of skins/themes available, both commercial and for free. There are also a lot of modifications (mods) available and many language files.

* Development Status: Production/Stable.
* License: GNU General Public License (GPL).
* Operating System: Cross Platform.
* Environment: Web Environment.
* Author: phpBB Group.
* Available Languages: Africaans, Albanian, Amharic, Arabic, Asturian, Azerbaijani, Basque, Belarusian, Bosnian, Breton, Bulgarian, Catalan, Chinese (Simplified), Chinese (Traditional), Croatian, Czech, Danish, Dutch, Esperanto, English, Estonian, Farsi, Finnish, French, Frisian, Galician, German, German (Formal), Greek, Hebrew, Hungarian, Icelandic, Indonesian, Italian, Japanese, Korean, Kurdish, Lithuanian, Macedonian, Mongolian, Mongolian (Latin), Norwegian (Bokmaal), Norwegian (Nynorsk), Polish, Portuguese, Portuguese (Brazil), Romanian, Russian, Sakha, Serbian, Serbian (Cyrillic), Slovakian, Slovenian, Spanish, Spanish (Argentina), Swedish, Tatarish, Thai, Turkish, Uighur, Ukrainian, Vietnamese, Welsh.
* Available Skins/Themes: -.
* Disk Space Required: - MB.
* Web Site: http://www.phpbb.com

PunBB

PunBB is a fast and lightweight PHP-powered discussion board. Its primary goals are to be faster, smaller and less graphically intensive as compared to other discussion boards. PunBB has fewer features than many other discussion boards, but is generally faster and outputs smaller, semantically correct XHTML-compliant pages.

PunBB is simple and fast, there is a rich community that develop different modifications (mods) to PunBB with which you can extend the functionality of PunBB. There is also a rich language support and you can change the look and feel with skins/themes (called styles by PunBB).

* Development Status: Production/Stable.
* License: GNU General Public License (GPL).
* Operating System: Cross Platform.
* Environment: Web Environment.
* Author: Rickard Andersson.
* Available Languages: Arabic, Bosnian, Bulgarian / Bulgarian (UTF-8), Catalan, Croatian (Hrvatski), Czech, Chinese (Simplified), Chinese (Traditional), Danish, Dutch, English, Estonian, Farsi, Finnish, French, German, Greek, Hebrew, Hungarian, Icelandic, Indonesian, Italian, Korean, Kurdish, Latvian, Lithuanian, Norwegian, Polish, Portuguese, Portuguese (Brazilian), Romanian, Russian, Serbian, Slovak, Slovenian, Spanish, Swedish, Turkish, Ukrainian, Vietnamese.
* Available Skins/Themes: -.
* Disk Space Required: - MB.
* Web Site: http://www.punbb.org

SMF

SMF (Simple Machines Forum)
SMF Logo.

Simple Machines Forum - SMF in short - is a free, professional grade software package that allows you to set up your own online community within minutes. Its powerful custom made template engine puts you in full control of the lay-out of your forum and with our unique SSI - or Server Side Includes - function you can let your forum and your website interact with each other.

SMF is designed to provide you with all the features you need while having an absolute minimal impact on the resources of the server. SMF is the next generation of forum software, and best of all it is and will always remain completely free!

You can change the look of your SMF-forum with skins/themes and add functions with modifications and hacks. SMF support many languages with language packs that can be downloaded from their web site.

* Development Status: Production/Stable.
* License: Other Free.
* Operating System: Cross Platform.
* Environment: Web Environment.
* Author: Simple Machines Team.
* Available Languages: Arabic, Brazilian (Portugese), Bulgarian, Catalan, Chinese (Simplified), Chinese (Traditional), Croatian, Danish, Dutch, English, Finnish, French, German, Greek, Hebrew, Hungarian, Norwegian, Persian, Polish, Portugese, Romanian, Russian, Spanish, Swedish, Thai, Turkish.
* Available Skins/Themes: -.
* Disk Space Required: - MB.
* Web Site: http://www.simplemachines.org

if there are others i missed please reply

thanks