Gorenstein stable surfaces with KX2=1 and pg > 0

In this paper we consider Gorenstein stable surfaces with K-X(2) = 1 and positive geometric genus. Extending classical results, we show that such surfaces admit a simple description as weighted complete intersection. We exhibit a wealth of surfaces of all possible Kodaira dimensions that occur as normalisations of Gorenstein stable surfaces with K-X(2) = 1; for p(g) = 2 this leads to a rough stratification of the moduli space. Explicit non-Gorenstein examples show that we need further techniques to understand all possible degenerations.