Most users encountering problems are seeing 37 and 12.
Error 37 can be translated to "The service is undergoing maintenance, please try again later." Error 12 means "the Global Play feature being disabled until all regions are active".
Please make sure you choose the correct region when starting the game as per this Error 0 - The service is undergoing maintenance, please try again later. Error 3004, 3006, 3007, or 300008 - There are a number of possible causes for these errors. Error 315300 - If you're sure you're typing your correct account information, check to make sure no additional spaces were added before or after your email address.
(Please note that Diablo 3 will not be available for play until May 15 at a.m. If you're still running into this issue, there may be an error in your foreign language appdata files.
During this time, however, it's possible that the error is caused by the number of people hitting the servers.
While attempting to play the D3 beta, I'm getting error 3004 when attempting to start a game.
I can't say it is what the folks at Blizzard would want/recommend (can't find an official response), but it is what others seem to be doing with some degree of success.
Possibly because the servers may able to support more players, but the login process maybe demands a bit more resources all at once, when compared to simply keeping you playing once already in-game?
Deleting the Cache Folder Instructions on deleting the Cache.
This is happening after choosing my character, and clicking resume game.
This error message is very succinct, but I can't help but want more information.
Though the downloader may show 100%, please allow some time for it to fully complete.
To resolve this issue, log into and re-run the downloader.