While there are plenty of decent open source bug trackers such as Bugzilla (as Navi mentioned) and Mantis Bug Tracker, I haven't found many which also include solid project management features like FogBugz and other licensed software.
I suppose that it also depends on what kind of project you are tracking. If it's something which doesn't require much more security than a username/password, you might find a good number of web services which can host your needs; however, if you work for a company/industry which requires more discretion (say- government?), then you most likely won't be able to take advantage of the greater web community for these needs.