How can a U.S. citizen open a bank account in Europe?
Tackling your last point, all banks in the EU should be covered to around €100,000. The exact figure varies slightly between countries, and generally only private deposits are covered. In the UK it's the FSCS that covers private deposits, to a value of £85,000, see this for more information on what's covered. In France (for a euro denominated example), there's coverage up to €100,000 provided by Fonds de Garantie des Dépôts, see this (in French) for full details. There's a fairly good Wikipedia Article that covers all this too. I'll let someone else chime in on the mechanics of opening something covered by the schemes though!
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How can a U.S. citizen open a bank account in Europe?

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Tackling your last point, all banks in the EU should be covered to around €100,000. The exact figure varies slightly between countries, and generally only private deposits are covered. In the UK it's the FSCS that covers private deposits, to a value of £85,000, see this for more information on what's covered. In France (for a euro denominated example), there's coverage up to €100,000 provided by Fonds de Garantie des Dépôts, see this (in French) for full details. There's a fairly good Wikipedia Article that covers all this too. I'll let someone else chime in on the mechanics of opening something covered by the schemes though!