Some twenty years ago there appeared, in the Annals of Mathematics, the marvelous memoir of Zariski entitled: Reduction of singularities of algebraic three-dimensional varieties. Not only was a daring and ingenious solution of a difficult problem given in it, but so much of the technique invented for the solution has proved to be of such significance for algebraic geometry in general! Hironaka's brilliantly energetic recent solution of the general resolution problem for zero characteristic constitutes, indeed, a high tribute to Zariski's memoir. At present I am able to pay only a modest tribute to Zariski's memoir by giving a self-contained exposition of it. This then is the primary aim of the monograph. A secondary aim is to partially extend some of the results to nonzero characteristic. The algorithm needed for such an extension has already been published in four papers, and it will not be repeated here. This monograph contains the geometric part of the argument. However, we do include an alternative simple version of the algorithm for zero characteristic thereby making the monograph self-contained for that case. Finally, the matter is so arranged that about half of the monograph can be used as an introduction to certain foundational aspects of algebraic geometry. [b]Be Happy!!![/b] [b] !!!No Mirrors below, please! Follow Rules! [/b]

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