Variety of styles, depending on the teacher's influences and or experience
American Mime Theatre was created in 1952.
An American Tail Theatre was created in 1991.
An American Tail Theatre ended in 1992.
American Ballet Theatre was created in 1956.
American Airlines Theatre was created in 2000.
American Shakespeare Theatre was created in 1955.
American Laboratory Theatre was created in 1923.
American Laboratory Theatre ended in 1933.
The American Place Theatre was created in 1963.
The American Ballet Theatre is a company so they perform. Once a dancer is employed, they usually do not need to be'taught' too much anymore. Its just a matter of keeping their skills at a high level, for use on stage.The school attached to the American Ballet Theatre is a different question. To the best of my knowledge they teach a combination of techniques that include; Vaganova, Chechetti, RAD and other European styles such as Royal Danish.
American Association of Community Theatre was created in 1986.
American Music Theatre Project was created in 2005.