  We study weakly first order cosmological phase transitions in finite
temperature field theories. Focusing on the standard electroweak theory and its
minimal supersymmetric extension, we identify the regimes of Higgs masses for
which the phase transition in these models proceeds by significant phase mixing
and the coarsening of the subsequent domain network. This dynamics is distinct
from that for strongly first order transitions, which proceed by the nucleation
and propagation of critical bubbles. We describe how electroweak baryogenesis
might take place in these models, explaining how our new picture can relax the
sphaleron washout bound of traditional scenarios.
