%0 Journal Article %T GIT Constructions of Log Canonical Models of M_g %A Jarod Alper %A Donghoon Hyeon %J Mathematics %D 2011 %I arXiv %X The purpose of this article is to give an overview of the construction of moduli spaces of curves from the viewpoint of the log minimal model program for M_g by providing an update of recent developments and discussing future problems. This survey is distinguished from the recent articles of Fedorchuk-Smyth and Morrison in its focus on low degree Hilbert stability of curves. %U http://arxiv.org/abs/1109.2173v1