  This largely expository paper first gives an introduction to Hilbert
stability and its use in Gieseker's GIT construction of $\overline{M}_g$. Then
I review recent work in this area--variants for unpointed curves that arise in
Hassett's log minimal model program, starting with Schubert's moduli space of
pseudostable curves, and constructions for weighted pointed stable curves and
for pointed stable maps due to Swinarski and to Baldwin and Swinarski
respectively. The focus is on the steps at which new ideas are needed. Finally,
I list open problems in the area, particularly some arising in the log minimal
model program that seem inaccessible to current techniques.
