A survey, thorough and timely, of the singularities of two-dimensional normal complex analytic varieties, the volume summarizes the results obtained since Hirzebruch's thesis (1953) and presents new contributions. First, the singularity is resolved and shown to be classified by its resolution; then, resolutions are classed by the use of spaces with nilpotents; finally, the spaces with nilpotents are determined by means of the local ring structure of the singularity.

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The detailed study of normal 2-dimensional singularities is much easier than the higher dimensional case primarily for two reasons. First, any normal 2-dimensional singularity p is isolated. Secondly, because p is 2-dimensional, A is 1-dimensional. The theory of compact Reimann surfaces gives a great deal of information about A and small neighborhoods of A.

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