  In this paper, we completely work out the log minimal model program for the
moduli space of stable curves of genus three. We employ a rational multiple
$\alpha\delta$ of the divisor $\delta$ of singular curves as the boundary
divisor, construct the log canonical model for the pair $(\bar{\mathcal M}_3,
\alpha\delta)$ using geometric invariant theory as we vary $\alpha$ from one to
zero, and give a modular interpretation of each log canonical model and the
birational maps between them. By using the modular description, we are able to
identify all but one log canonical models with existing compactifications of
$M_3$, some new and others classical, while the exception gives a new modular
compactification of $M_3$.
