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In this paper we consider Gorenstein stable surfaces with $K^2_X=1$ andpositive geometric genus. Extending classical results, we show that suchsurfaces admit a simple description as weighted complete intersection.  We exhibit a wealth of surfaces of all possible Kodaira dimensions that occuras 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 tounderstand all possible degenerations.

Gorenstein stable surfaces with $K_X^2 = 1$ and $p_g>0$