This article gives a classification scheme of algebraic transformations of Gauss hypergeometric functions, or pull-back transformations between hypergeometric differential equations. The classification recovers the classical transformations of degree 2, 3, 4, 6, and finds other transformations of some special classes of the Gauss hypergeometric function. The other transformations are considered more thoroughly in a series of supplementing articles.

Comment: 29 pages; 3 tables; Uniqueness claims and Remark 7.1 clarified by footnotes; formulas (28), (29) corrected