Can I live and work in Germany without a Visa or extended permission to stay?
Citicens of EU countries can live and/or work in Germany without Visa, work permit or permission to stay. US citicens (and citicens of many other countries) can enter Germany (and every other EU country) for not more than three months without a Visa. For longer stays you’ll need to apply for a permission to stay and a work permit. This would be different if you were married to a German citizen. Then you’d be entitled for a stay and work permit, limited for two years in the beginning and permanent after these two years. BTW: The permanent permit would not be revoked in case the marriage gets divorced.
