The answer is: IT DEPENDS ON THE BANK!
And the country.
Many, many banks are only used to dealing with depositors from their own country.
They may not even be able to handle street addresses or phone numbers which are not in their own country.
Or worse, they may appear to handle it fine, but then not put the right stamp on the envelope when they send you your statements, which you will never get :)
Many countries have strict rules (like the US PATRIOT rules) that make it almost impossible to take deposits from people outside the country.
I would go so far as to say that it's going to be incredibly difficult if not impossible to open a US bank account without physically going to the bank, AND having a permanent mailing address in the US.
That said, some banks (and countries) are particularly good at this.
There are "offshore" banks in Jersey, the Isle of Man, the Cayman Islands, etc., which have been specifically set up for the purpose of allowing depositors from anywhere in the world to open accounts.
If you search Google for "Offshore Banking" you'll find a lot of references and links to banks that are set up to do this.
money.stackexchange.com/users/876/Joel Spolsky
I don't think its as hard as you might think.
I personally know of someone who set up an account at "Loyal Bank"
and I know they didn't have to leave the country to do it: https://www.loyalbank.com
I just went there and clicked on "Open Personal Account" and basically it looks like you just have to print and mail them signed documents and proof of identification, etc.
Pretty easy stuff...
money.stackexchange.com/users/251/TaxQueries.com
Opening a bank account in Canada as a US citizen requires you to physically go to Canada.
I'm not sure about other countries.
money.stackexchange.com/users/1229/Muro
I opened an account with saxobank.com (Denmark) w/ no problem.
Filled out online forms, scanned my driver's license and 1-2 other documents, wired them money, and was good to go.
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I have worked in France and had to open a bank account there.
Boy, the amount of administration that goes into that...
But above all, you have to prove that you reside in France to be able to open the account.
The catch: to reside in France, you need to have a bank account.
I can't imagine that you can open an internet account in a French bank with that complicated a system.
money.stackexchange.com/users/1930/Raskolnikov
Your question isn't clear about which countries are "foreign", but it's basically impossible to open a USA bank account if you are not a citizen or resident.
money.stackexchange.com/users/1175/user1175
Has anyone attempted opening a foreign bank account from Nepal as a Nepalese citizen, if so what was your experience? how easy was it? Which country's banks will allow this?
How easy is it for people from Nepal to open a foreign bank account via the internet?
money.stackexchange.com
<banking><online-banking><nepal>
money.stackexchange.com/questions/913
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It is impossible to open a US bank account unless you are a citizen or resident. Other countries might allow it, such as Denmark, but it depends on the bank. The answer will depend on the bank and the country in question. Some countries make it very difficult, if not impossible, to open a bank account without being physically in the country. Other countries make it a little easier.
In some countries, it is physically impossible to open a bank account without being in the country. In some countries, it is fairly easy to open a bank account without being physically present. The answer will depend on the bank.