There are no requirements for a COVID test for domestic travel within the United States. Internationally, 68 countries have lifted all entry restrictions for travelers arriving from the United States regardless of vaccination status and another 66 countries allow entry without a test provided you show proof of vaccination. When returning to the United States, airline passengers are no longer required to take a pre-travel COVID test. (More on that in the answer to Question 5.)
The U.S. Department of State offers country-specific travel information here.
Among the countries that have lifted all restrictions for arrivals from the U.S. are: Argentina, Aruba, Austria, Belgium, Costa Rica, Croatia, Curaçao, Cyprus, Czech Republic, Denmark, Egypt, El Salvador, Germany, Greece, Greenland, Grenada, Guam, Hungary, Iceland, Ireland, Italy, Jamaica, Maldives, Mexico, Norway, Poland, Puerto Rico, Romania, Saudi Arabia, South Africa, Sweden, Switzerland, Turkey, U.S. Virgin Islands and the United Kingdom.
Countries requiring vaccination but no COVID-19 test for arrivals from the U.S. include: Australia, Bahamas, Belize, Brazil, Cambodia, Canada, Cook Islands, Dominican Republic, Finland, France, Kenya, Luxembourg, Malawi, Malaysia, Malta, Monaco, Morocco, Mozambique, Netherlands, Nigeria, Pakistan, Panama, Paraguay, Peru, Philippines, Saint Martin, Senegal, Singapore, Spain, Thailand, Turks and Caicos, United Arab Emirates and Uruguay .