Free

Agstools
BugTrack, simple free bugtracking system based on perl5 and any DBI compliant database.
http://www.agstools.com/

Bug-A-Boo
An open source bug tracking system running on any web server allowing the execution of tdbengine binaries. It is dristributed unter the terms of the GPL.
http://www.bug-a-boo.org/

Bug-Buddy
A graphical bug reporting utility for GNOME. It allows users to easily make quality bug reports.
http://directory.fsf.org/bugbuddy.html

Bugs Online
A free (GPL) bug tracking system for use with Microsoft IIS, Access and ASP.
http://bugsonline.sourceforge.net/

BugTracker.NET
Opensource web-based bug tracker written using ASP.NET and C#.
http://btnet.sourceforge.net

Bugzilla
Bug tracking used by the Mozilla projects. Inherently web-based, written in Perl, and uses MySQL as its database back-end. Open-Source. Updates available via CVS.
http://www.bugzilla.org/

Debian Bug Tracking Software
GPL'd Perl-5 scripts which maintain a database of problem reports. Input by email, listings via webserver.
http://www.chiark.greenend.org.uk/~ian/debbugs/

GIFFT
Web based bug tracking system running on Windows, Linux, Unix as a CGI for Apache or other web server.
http://bkiss.free.fr/Gifft/

IssueTrackerProduct
Open-source web-based issue/bug tracker, written in Python.
http://www.issuetrackerproduct.com/

JitterBug
A web based bug tracking system, originally developed by the Samba Team to handle bug tracking, problem reports and queries, now available under the GNU General Public License. Operates like a communal Web based email system - no database back end required.
http://samba.anu.edu.au/cgi-bin/jitterbug

Keystone
Data stored in a SQL database, accessed by any Web browser which understands forms, tables, and cookies. By StoneKeep consulting. Free for under 10 users.
http://www.stonekeep.com/

Mantis
php/MySQL/web based bugtracking system.
http://mantisbt.sourceforge.net/

phpBugTracker
A web-based bug tracking system built in PHP.
http://phpbt.sourceforge.net/

Roundup
A simple-to-use and -install issue-tracking system with command-line, web and e-mail interfaces.
http://roundup.sourceforge.net/

RT: Request Tracker
A free web and email-based bug tracking and trouble ticketing system. Features list, documentation screen shots, and download.
http://www.bestpractical.com/rt/

Teamatic
Free web-based bug tracking, ticketing and project resource tracking system. Designed to meet the requirements of most Information Technology, Consulting and Engineering projects.
http://www.teamatic.com/

Trac
Web-based software project management and bug/issue tracking system. Provides an interface to Subversion and an integrated wiki.
http://trac.edgewall.org/

WREQ
Work Request, a distributed request/problem tracking system, written in Perl 5. Access by Web or email.
http://www.math.duke.edu/~yu/wreq/