It seems like a lot of people here are having trouble getting this game to run as of a day after release. I'm getting tired of seeing a ton of threads about the same things, so here we go.
First, download the 1.1 patch.
Punkbuster Problems: Download PBSetup.exe and find the game path, add game, and update. Allow punkbuster and World at War through your firewall(s).
Black Screen Problems: Update Directx 9 from Microsoft's Web Site. If this doesn't work, right click the exe (at least in vista) to run as an administrator. Lastly, alt tab out of the game to the desktop, and go back into the game.
I'll update this as I find out more.