The Central Limit Theorem (in the classical sense) is the generic name of a class of theorems which give, in precise mathematical terms, conditions under which the distribution function of a suitably standardized sum of independent random variables is approximately normal. This theorem is one of the most remarkable results in all of mathematics and is a dominating personality in the world of probability and statistics.
Adams, William J. The Life and Times of the Central Limit Theorem, Kaedon Publishing Company 1974, pp. 1-2.