Introduction. The reduction of the singularities of an algebraic surface has been carried out by B. Levi, Chisini, and Albanese.' These reductions are all of an essentially geometric nature, the first two using Cremona transformations of S3 while the last depends on the properties of linear systems of curves on the surface. In the present paper we propose to give an analytic method of reduction, that is, to construct the polynomials which define the transformation by making use of the properties of analytic functions of one or more variables. There is a two-fold reason for doing this. In the first place, for a theorem which is of such fundamental importance in the theory of surfaces it is desirable to have a completely rigorous proof which makes use of as few as possible of the peculiar properties of algebraic surfaces. Secondly, what is more important, it is hoped that the methods used here can be extended to varieties of more than two dimensions; up to the present time the advances in this direction have been negligible. In order to make full use of the simpler properties of analytic functions it is desirable to restrict our considerations as much as possible to neighborhoods of points on our.surface. In attempting to do this we are brought into contact with another problem, that of parametrizing the neighborhood of a point on an analytic surface. This problem has been completely solved by Black and by Jung,2 and in Part IV we shall include a simplified version of the most important part of Jung's proof. Part I is devoted to the formulation of the problem and its reduction to essentially local considerations. This is done by introducing the notions of parametrized wedges and their reducing systems. The rest of the paper is then concerned with the problem of constructing wedges and reducing systems of the required types for a neighborhood of an arbitrary point P of the surface. In Part II we consider the trivial case where P is non-singular, and then show