Last we've heard, Captain Janeway destroyed their Queen and their Unicomplex in the Voyager finale, "Endgame". Of course, this doesn't necessarily mean that they're gone for good; a good portion of the collective is probably still largely intact and even if their link is severed, could possibly put themselves back together. Long story short, the Borg case is not closed yet.
That, of course, is if you go by true Star Trek canon. In the Star Trek: Destiny trilogy of books published in late 2008, the Borg are destroyed by a race called the Celiar, the same species that accidently created the Borg 6,000 years ago.
Then there's Star Trek: Online. The Borg are too great an adversery to get rid of like in Star Trek: Destiny. They're back and as annoying as ever in the 2409 based MMO Star Trek Online. They've also made some more advancements and has more of the Queen's puppets in it (aka Borg drones like Locutus of Borg who have slight individuality but not enough to defy the Queen's orders).