  Standard perturbative (or mean field theory) techniques are not adequate for
studying the finite-temperature electroweak phase transition in some cases of
interest to scenarios for electroweak baryogenesis. We instead study the
properties of this transition using the renormalization group and the
$\epsilon$-expansion. This expansion, based on dimensional continuation from 3
to $4{-}\epsilon$ spatial dimensions, provides a systematic approximation for
computing the effects of (near)-critical fluctuations. The $\epsilon$-expansion
is known to predict a first-order transition in Higgs theories, even for heavy
Higgs boson masses. The validity of this conclusion in the standard model is
examined in detail. A variety of physical quantities are computed at leading
and next-to-leading order in $\epsilon$. For moderately light Higgs masses
(below 100~GeV), the $\epsilon$-expansion suggests that the transition is more
strongly first order than is predicted by the conventional analysis based on
the one-loop (ring-improved) effective potential. Nevertheless, the rate of
baryon non-conservation after the transition is found to be {\em larger\/} than
that given by the one-loop effective potential calculation. Detailed
next-to-leading order calculations of some sample quantities suggests that the
$\eps$-expansion is reasonably well behaved for Higgs masses below 100--200
GeV. We also compare the $\eps$-expansion with large-$N$ results (where $N$ is
the number of scalar fields) and find that the $\eps$-expansion is less well
behaved in this limit.
