The Diablo III login screen displays a Battle.net Notice acknowledging the Error 37 issue. Basically, keep trying every 5/10 minutes until you are lucky and get through the avalanche of login attempts. Blizzard IT Team is working on it.

Welcome to Diablo III! The fight for Sanctuary has begun.

Please note that due to a high volume of traffic, login and character creation may be slower than normal. If you’re unable to login to the game (Error 37) or create characters, please wait and try again. We’ve temporarily taken our Battle.net websites offline and launched more servers to accommodate for the traffic. We hope to resolve these issues as soon as possible and appreciate your patience.