Stable surfaces can be regarded as a generalization of stable curves: they have "well behaved" singularities (e.g., they are at most nodal in codimension 1) and, by the theory developed by Kollar, Shepherd Barron and Alexeev, their moduli space is complete, and thus it provides a compactification of the moduli space of surfaces of general type. I will report on work in progress with M. Franciosi and S. Rollenske on the fine classification of Gorenstein stable surface with K²=1, namely the smallest possible value. Our work combines suitable generalizations of the classical technique of studying the canonical ring and the pluricanonical maps with the point view introduced by Kollar of regarding stable surfaces as the result of glung a normal surface to itself along a certain divisor.