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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In algebraic geometry, a regular map between affine varieties is a mapping which is given by polynomials. For example, if X and Y are subvarieties of An resp. Am, then a regular map from X to Y is given by m polynomials in the n coordinates of An. More generally, a map :X Y between two varieties is regular at a point x if there is a neighbourhood U of x and a neighbourhood V of (x) such that the restricted function :U V is regular. Then is called regular, if it is regular at all points of X.