It's not a traditional game, and would never have had a traditional "release date", so today is the closest DOTA 2 will get. After years of betas, invites, sign-ups and queues, the game is now open to everyone.
If you visit the game's Steam page, you're guaranteed for the first time to see the green "play game" button, regardless of who you are, when you're playing or where you're at.
Why the long wait? Valve's been very careful to ensure it's got the infrastructure to cope with a large playerbase, slowly ramping up the number of gamers alongside its ability to handle them.
Example: even when the game first shed its beta label in July, Valve had this to say:
If you want to give Dota a try, or you have a friend you want to bring into the community, click the button above and you can get in the launch queue.
Today, you don't need to bother. Just play the game already.
Valve says there are currently 6.5 million people playing DOTA 2. A number that's now only going to get bigger.