The largest bank in Iceland, Kaupthing, was seized by the government in early October 2008. Kaupthing and several other major banks, including Landsbanki and Glinir, were caught in an economic crisis with the Icelandic government. It seemed that Iceland was going bankrupt. With pleas to Russia asking for $4 billion in aid, and with the help of the United Kingdom storing money for a rainy day, Iceland seem to pull out their crisis. Now, Iceland guarantees Kaupthing that any money stored within their vaults are protected. According to the Kaupthing website, the Icelandic government states that deposits in retail banks are fully guaranteed. If you are satisfied your money will be safe, here's how to open a bank account with an Iceland bank
Show the bank your Icelandic ID number. This ID number must be attained through the National Registry (thjodskra).
Show the bank your passport. That's it. The passport and Icelandic ID number are all the documentation that is required to open a banking account in Iceland.
Give the bank a deposit. The amount of the deposit varies greatly for each bank
Create a bank account. Once you have completed steps 1 through 3 you can create an account. A typical account will consist of a debit card, online banking, a savings account and a basic checking account. If you are working in Iceland, the bank will offer a pension plan as well.
Kaupthing, Glitner, and Landsbanki were all government banks. They all became privatized. Now they are again under government control since the Icelandic financial crisis of 2008. All three banks are guaranteed by the government that your money is safe.