Classical and minimal models of the moduli space of curves of genus two

This semi-expository paper discusses the log minimal model program as applied to the moduli space of curves, especially in the case of curves of genus two. Log canonical models for these moduli spaces can often be constructed using the techniques of Geometric Invariant Theory. In genus two, this boils down to the invariant theory of binary sextics, which was developed systematically in the 19th century.