How can an American move to Monaco?

The practical sequence is: secure accommodation in Monaco (rental lease or property deed), open an account at a Monaco-licensed bank and meet its minimum deposit (typically €500,000 to €1,000,000+), apply for a French long-stay (type D) visa at the French consulate covering your US state of residence, travel to Monaco on that visa, then apply for a Monaco carte de séjour at the Direction de la Sûreté Publique. You will need to provide proof of accommodation, the bank reference letter, a long-form birth certificate (apostilled), a criminal record check covering the last five years from each country lived in, and proof of health insurance. The process typically takes 6 to 12 months end-to-end. After three annual renewals the card converts to a three-year version, and after ten years you can apply for the privileged ordinaire (long-term) status. For the full cost picture, see our Monaco cost of living page.