For countries in the EU, Americans can stay 90 days without a visa of any kind. In the U.K., Americans can stay up to 6 months visa-free. You are not permitted to work in either situation. If you are a digital nomad hoping to move to Europe long-term, many countries in Europe now offer a digital nomad visa. Portugal was one of the first countries to offer this, but many other countries have followed suit: like Spain, Croatia and Montenegro.