I will start surveying recent results on Chern slope geography (joint with X. Roulleau in char=0, with R. Codorniu in char>0), then pass to some results (joint with J. Rana, J. Tevelev; and joint with S. Coughlan) on p_g=0 surfaces, and open questions in relation to their moduli space. The Lee-Park connection between moduli spaces and geography will bring into the picture the explicit MMP of Q-Gorenstein degenerations (joint with P. Hacking and J. Tevelev), and the possible Wahl singularities of singular surfaces in the KSBA boundary (joint with A. Stern), in particular I will discuss on sharp bounds on the indices of these singularities for a fixed K^2 (joint with J. Rana). I will explain results, optimal examples, and open problems on that. Speaker(s): Giancarlo Urzua (Pontificia Universidad Catolica de Chile)