Bank accounts & banking in Rome
Most Italian banks won't let you open an account unless you have residential status. However, this can vary, and you hear conflicting reports of what banks can and can't do for you. It can depend on the bank, the branch, the assistant, the weather...
Some people have also been able to open their Italian bank accounts (at least provisionally) by visiting the branch in their native country. Several Italian banks have additional branches around the world, so this can be worth looking into. Finally, if you have trouble opening a bank account, you may be able to open a post office account instead.
