  We study the dynamics of phase transitions out of equilibrium in weakly
coupled scalar field theories. We consider the case in which there is a rapid
supercooling from an initial symmetric phase in thermal equilibrium at
temperature $T_i>T_c$ to a final state at low temperature $T_f \approx 0$. In
particular we study the formation and growth of correlated domains out of
equilibrium. It is shown that the dynamics of the process of domain formation
and growth (spinodal decomposition) cannot be studied in perturbation theory,
and a non-perturbative self-consistent Hartree approximation is used to study
the long time evolution. We find in weakly coupled theories that the size of
domains grow at long times as $\xi_D(t) \approx \sqrt{t\xi(0)}$. For very
weakly coupled theories, 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.
