I can't thank "Greyhead" enough for providing this great component for years.
I've managed forums for years, and must admit Kunena is still a bit behind other forum stand alone platforms, like phpBB or SMF, but it's making the gap narrower with every new release, and it offers something any of these platforms can't: fully native integration. Also, it's not a "new" project, as it comes from a previous one, Fireboard, which also came from an older one... so there's a lot of "forum wisdom" in this project.
It works together with many other Joomla extensions (Joomla Articles, UddeIM, Community Builder, JomSocial, Easy Social), and allows you creating interactive sites, or directly community sites.
So, if loooking for a forum for a Joomla site, check Kunena first. You won't regret.
Simply put, after using it for a few months you'll loose the count of how many times it saved your life, or how much time it saved you for moving sites from one server to another.
And the most amazing thing: it's free! I'm sure many people in the software industry won't be able to believe it :).