The 1983 film about a determined Cuban immigrant who takes over a drug empire is Scarface.
Directed by Brian De Palma and written by Oliver Stone, Scarface stars Al Pacino as Tony Montana, a Cuban refugee who arrives in Miami during the 1980 Mariel boatlift. Tony begins with little money and few opportunities, then works his way into Miami’s cocaine trade. After turning against crime boss Frank Lopez, he builds his own organization and becomes one of the city’s most powerful drug dealers.
Scarface is a remake of Howard Hawks’s 1932 film of the same name, which focused on an Italian American gangster during Prohibition. The remake shifted the story to contemporary Miami and the cocaine trade. Its extreme violence and portrayal of Cuban immigrants sparked controversy, but the film later became a cult classic and a major influence on popular culture, especially hip-hop and crime-themed video games.
A common mix-up is confusing this film with the 1932 Scarface. The question’s 1983 date identifies Pacino’s version.