The United States has a long and complex history of military intervention and invasions in other countries. The number of states that have been invaded by the United States varies depending on the definition of "invasion." Some experts consider any use of force, no matter how limited, to be an invasion, while others reserve the term for larger-scale military operations.

A comprehensive list of all the countries that the United States has invaded is difficult to compile, as there have been so many instances of military intervention, ranging from minor skirmishes to full-scale invasions. The following is a list of some of the most significant invasions in U.S. history:

1. Mexico (1846-1848): The Mexican-American War was fought between the United States and Mexico over territorial disputes. The U.S. military invaded Mexico and captured Mexico City, forcing Mexico to sign the Treaty of Guadalupe Hidalgo, which ceded large territories to the United States, including what is now California, Nevada, Utah, Arizona, New Mexico, and parts of Colorado, Wyoming, Kansas, and Oklahoma.
2. Philippines (1898-1902): The U.S. invaded the Philippines during the Spanish-American War and subsequently fought a guerilla war against Filipino nationalists who sought independence from the United States.
3. Vietnam (1955-1975): The United States intervened in the Vietnam War to prevent the spread of communism in Southeast Asia. The U.S. military conducted large-scale ground operations and air strikes against North Vietnam, as well as fighting a guerrilla war against the Viet Cong in South Vietnam. The war ended in 1975 with the fall of Saigon to communist forces.
4. Iraq (2003-2011): The United States invaded Iraq in 2003, citing concerns over Iraq's alleged possession of weapons of mass destruction and ties to terrorist groups. The U.S. military ousted Iraqi dictator Saddam Hussein, but the war resulted in a prolonged insurgency and sectarian violence.
5. Afghanistan (2001-2021): The U.S. invaded Afghanistan in 2001 in response to the 9/11 terrorist attacks. The U.S. military has been engaged in a protracted conflict against the Taliban and other insurgent groups in Afghanistan ever since.

In addition to these invasions, the United States has engaged in numerous other military interventions and smaller-scale operations in other countries throughout its history. Some of these interventions have been controversial and have sparked debate over the role of the United States in the world.

It is worth noting that not all military interventions by the United States have been invasions, and not all invasions have been outright military conquests. Some interventions have been conducted for humanitarian reasons, while others have been aimed at supporting friendly governments or protecting American interests abroad. Nonetheless, the history of U.S. military intervention in other countries is a complex and controversial topic that continues to be debated by scholars, policymakers, and the public.