  In this paper we consider Gorenstein stable surfaces with $K^2_X=1$ 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.
