I will briefly recall the notion of stable surfaces and of the corresponding moduli space. Then I will outline a partial description of the boundary points in the case of surfaces with $K^2=1$, $p_g=2$ (joint work with Stephen Coughlan, Marco Franciosi, Julie Rana and Soenke Rollenske, in various combinations) and, time permitting, in the case of Campedelli and Burniat surfaces (joint work with Valery Alexeev).