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The organization of faculties into departments is not standardized, but most U.S. universities will have at least departments of History, Physics, English (language and literature), Psychology, and so on. Sometimes divisions are coarser: a liberal arts college which de-emphasizes the sciences may have a single Science department; an engineering university may have one department for Language and Literature (in all languages). Sometimes divisions may be finer: for example, Harvard University has separate departments of Organismic and Evolutionary Biology, Molecular and Cellular Biology, and Chemistry and Chemical Biology. Some disciplines are found in different departments at different institutions: biochemistry may be in biology, in chemistry or in its own department; computer science may be in mathematics, applied mathematics, electrical engineering, or its own department (the usual case nowadays).
Departments are generally chaired by a member of the department, who may be elected by the faculty of the department, appointed by the dean of the faculty, or assigned by simple rotation among the tenured faculty. The duties, importance, and power of the department chair vary widely among institutions and even among departments within an institution.
Courses are generally given within a department, and often named for the department, e.g. Physics 230: Quantum mechanics.
Graduate students in academic (as opposed to professional) programs are much more closely tied to their departments than undergraduates, and the department, rather than the university, is almost completely responsible for their selection (cf. college admissions) and course of study.