  These lectures address the dynamics of phase ordering out of equilibrium in
condensed matter and in quantum field theory in cosmological settings,
emphasizing their similarities and differences. In condensed matter we describe
the phenomenological approach based on the Time Dependent Ginzburg-Landau
(TDGL) description. After a general discussion of the main experimental and
theoretical features of phase ordering kinetics and the description of linear
(spinodal) instabilities we introduce the scaling hypothesis and show how a
dynamical correlation length emerges in the large N limit in condensed matter
systems. The large N approximation is a powerful tool in quantum field theory
that allows the study of non-perturbative phenomena in a consistent manner. We
study the exact solution to the dynamics after a quench in this limit in
Minkowski space time and in radiation dominated Friedman-Robertson-Walker
Cosmology. There are some remarkable similarities between these very different
settings such as the emergence of a scaling regime and of a dynamical
correlation length at late times that describe the formation and growth of
ordered regions. In quantum field theory and cosmology this length scale is
constrained by causality and its growth in time is also associated with
coarsening and the onset of a condensate. We provide a density matrix
interpretation of the formation of defects and the classicalization of quantum
fluctuations.
