  We advocate a (Wilson) renormalization-group (RG) treatment of
finite-temperature first-order phase transitions, in particular those driven by
radiative corrections such as occur in the standard model, and other
spontaneously-broken gauge theories. We introduce the scale-dependent
coarse-grained free energy $S_\La[\phi]$ which we explicitly calculate, using
the Wilson RG and a $(4-\ep)$-expansion, for a scalar toy model that shares
many features of the gauged case. As argued by Langer and others, the dynamics
of the phase transition are described by $S_\La[\phi]$ with $\La$ of order the
bubble wall thickness, and {\it not} by the usual (RG-improved)
finite-temperature effective action which is reproduced by $S_\La[\phi]$ for
$\La\to 0$. We argue that for weakly first-order transitions (such as that in
the standard model) the $(4-\ep)$-expansion is necessary to control an
inevitable growth of the effective scale-dependent coupling towards the
strong-coupling regime, and that diagrammatic resummation techniques are
unlikely to be appropriate.
