Final answer:
Emperor Maximilian I and Empress Carlota were invited by Napoleon III of France to rule Mexico.
Step-by-step explanation:
The French monarchs who were invited to rule Mexico were Emperor Maximilian I and Empress Carlota. This invitation came from Napoleon III of France in the 1860s.
Napoleon III saw an opportunity to establish a pro-French monarchy in Mexico while the United States was preoccupied with its own Civil War.
France initially captured Veracruz and moved to capture Mexico City, but Mexican forces ultimately defeated the French in 1862.
Despite this setback, France eventually conquered Mexico and installed Maximilian I as the emperor. However, the French rule in Mexico was short-lived, and Maximilian I was executed in 1867.