1. In [10] (p. 650) we have proved a uniformization theorem for zero-dimensional valuations on an algebraic surface, over an algebraically closed ground field K (of characteristic zero). In the present paper we generalize this theorem to algebraic varieties, and on the basis of this generalization we obtain a solution of the problem of local uniformization in the classical case (i.e. when K is the field of complex numbers). The exact formulation of the generalized theorem, in its strongest form, will be given in A III and A IV. However, to begin with, we state here the following theorem which is literally a repetition of our theorem for surfaces, with the surface replaced by a variety, and which will be included in our final result: THEQREM U1. The Uniformization Theorem in invariantive form: Given a field 2 of algebraic functions of r independent variables, over an algebraically closed ground field K of characteristic zero, and given a zero-dimensional valuation B of 2, there exists a projective model V of 2 on which the center of B is at a simple point P. This theorem is in effect entirely invariantive in nature: it refers exclusively to the field 2 and to the valuation B of 2. It asserts the existence of uni-