We'll keep this post updated and time-stamped with the latest news. You'll only need to look here to know.
6:45 p.m. EST: There is still no ETA on the fix. Downtime will continue for the time being. Another update will be provided in an hour. In addition to the game being down, the forums are down, the Blizzard store appears to be having issues, and the official Warcraft website is loading extremely slow. WoW.com is handling the excess traffic fine, thankfully, so we should be able to continue to provide you with any updates we get.
7:05 p.m. EST: We are getting reports that some realms are up. Your mileage may vary.
Older updates are after the break.
2:13 p.m. EST: Maintenance extended, update in one hour. (source)
3:10 p.m. EST: Realms are still down, and the official WoW forums appear to be down as well. Expect some more downtime.
3:25 p.m. EST: Still no ETA on having the realms up, but according to the Warcraft twitter they will provide another update in an hour.
4:00 p.m. EST: Many realms are coming up online now.
4:03 p.m. EST: According to the Warcraft twitter account, all realms should be playable within the next hour.
4:13 p.m. EST: Many many players are now reporting being stuck at the loading screen. The loading bar will get to the end, and then their computer will just hang there. No word from Blizzard yet.
4:15 p.m. EST: Blizzard has confirmed there is a problem which is keeping players stuck in the loading screen. No fix for it at the moment. It looks like it's an issue for everyone, although we don't know for certain yet.
4:47 p.m. EST: Most (all?) Players are now getting stuck at connected.
5:10 p.m. EST: There is still no resolution, or any update to give.
5:15 p.m. EST: It seems that now some people can connect, but are being disconnected as soon as they go to their realm.
5:45 p.m. EST: According to the Warcraft twitter there is still no resolution to the problem. Blizzard will provide another update in an hour. Some people are able to log onto the realms, but immediately get disconnected again.
6:15 p.m. EST: We've received dozens of reports that the Blizzard store is having issues selling the authenticators. Looks like you might have to wait awhile to get one if you're buying it to get the in-game core hound pup.