Nothing nowadays but in the past you could bank money etc without even giving your name, they would give you a number and that's it. Switzerland bacame the place to stash all sorts of dodgy dosh.
Swiss banks aren't what they used to be. They used to guarantee complete anonymity, ingrained in law. It's changed now though, and the Government can force the bank to give them account holder's details.
I believe Luxembourg still has banks that sdhere to the old Swiss code.