  The dynamics of typical phase transitions is studied out of equilibrium in
weakly coupled inflaton-type scalar field theories in Minkowski space. The
shortcomings of the effective potential and equilibrium descriptions are
pointed out. A case of a rapid supercooling from $T_i>T_c$ to $T_f \ll T_c$ is
considered.
  The equations of motion up to one-loop for the order parameter are obtained
and integrated for the case of ``slow rollover initial conditions''. It is
shown that the instabilities responsible for the process of phase separation
introduce dramatic corrections to the evolution. Domain formation and growth
(spinodal decomposition) is studied in a non-perturbative self-consistent
approximation.
  For very weakly coupled theories domains grow for a long time, their final
size is several times the zero temperature correlation length. For strongly
coupled theories the final size of the domains is comparable to the zero
temperature correlation length and the transition proceeds faster. We also
obtain the evolution equations for the order parameter and the fluctuations to
one-loop order and in a non-perturbative Hartree approximation in spatially
flat FRW cosmologies. The renormalization, and leading behavior of the high
temperature limit are analyzed.
