You want (or need) to host the solution yourself or not?
While not free or open source, I heartily recommend
Jira. The other projects mentioned by wraith808 are very good and mature issue-trackers, but the interface and manageability of Jira is awesome. They have a cheap plan of 10 USD/month for a max of 10 users and prices go down if you need to add more people. There is also their solution that you can host on your own hardware.
In any case, Jira is already very feature complete and if you put in the time, you can manage every aspect of the workflow you like or need. And if things are still missing, you can make your own plugin(s) or get one from their shop. Which is mainly filled with plug-ins that other 3rd-party companies have made and they offer these either freely or with a commercial license.
Jira is a product from the Atlassian company and integrates very well with other Atlassian products. Been using Jira (self-hosted) for more than 10 years already and it is really an excellent tool for keeping track of issues, it integrates very well with 3rd party version tracking of files out-of-the-box, keeps track of time spent on issues, generating reports, graphical overviews and lots more. Never had one inkling of using anything else since.
Atlassian software, that truly is money well spent. No affiliation, besides being a happy user.
