Mantis Bug Tracker is a free and open source, web-based bug tracking system. It is written in the PHP scripting language and works with MySQL, MS SQL, and PostgreSQL databases and a webserver. MantisBT can be installed on Windows, Linux, Mac OS, OS/2, and others. Almost any web browser should be able to function as a client.
Moreover 
MantisHub is available as commercial hosted solution of MantisBT.
Supports any platform that runs PHP (Windows, Linux, Mac, Solaris, AS400/i5, etc)
Available in 68 localizations.
Customizable Issue Pages
Multiple Projects per instance
Support for Projects, Sub-Projects, and Categories.
Users can have a different access level per project
Changelog Support
Roadmap
My View Page
Search and Filter
Full Text Search
Simple/Advanced Filters
Shared Filters (across users / projects)
Built-in Reporting (reports / graphs)
Custom Fields
Email notifications
Users can monitor specific issues
Attachments (can be saved on webserver or in database - can also backup to an FTP account)
Issue Change History
RSS Feeds (news, issues matching saved filters, issues matching a specific project)
Customizable issue workflow
Sponsorships Support - users are able to place bounties or sponsorships for specific issues, also developers can track such sponsorships / payments.
Anonymous Access
Signup with Captcha and Email Confirmation.
Self-Service Reset Password Support
Export to csv, Microsoft Excel, Microsoft Word
Ability to extended functionality through hook (custom) functions.
Ability to extended functionality through plugins.
Reporting issues via Email (available as a patch for 1.1.x and EmailReporting Plugin for 1.2.x]])
Reporting Issues via a custom form on your website (Anonymantis).
Source Control Integration (GIT, SVN and CVS).
No limit on the number of users, issues, or projects.
Wiki Integration (optional)
Time Tracking
Ability to send messages to messages to other users in regards to a specific issue.
kamalF added Mantis Bug Tracker as alternative to Adore
sheridanbulger added Mantis Bug Tracker as alternative to Essembi
Comments and Reviews
No way to login with Google and such...
This software hasn't been really fixed or improved upon for over a decade, save yourself the pain and just find some other tracker. I decided to give this software another shot: I installed it and then immediately found that I couldn't create users - you can CREATE users, but you can't give them a password from the administrator console, which seems ludicrous. The default PDO is the depreciated mysql (not mysqli, although that is an option) I needed to create about 20 users and saw that this was pretty much impossible to do easily, so I gave up and looked to find something better.
Pro: There don't seem to be many free bugtrackers out there but MantisBT does a great job once you get it set up correctly and spend hours of time exploring its capabilities. The bug lists are clear and neat.
Contra: I spend a lot of time going back and forth between admin pages to find what i am looking for, which can be a bit aggravating at times.
I upgraded the software once, only to find there were a ton of error messages popping up and those backups came in handy. It only seems to be getting worse with newer releases, so i've decided to stop trying. It's a shame though, i like MantisBT, i really do! Maybe i'm just doing something wrong, who will tell?