  The physics during the inflationary stage of the universe is of quantum
nature involving extremely high energy densities. Moreover, it is out of
equilibrium on a fastly expanding dynamical geometry.We present in these
lectures non-perturbative out of equilibrium field theoretical methods in
cosmological universes. We then study the non-linear dynamics of quantum fields
in matter and radiation dominated FRW and de Sitter universes. We investigate
the explosive particle production due to spinodal instabilities and parametric
amplification in FRW and deSitter universes with and without symmetry breaking.
We show how the particle production is sensitive to the expansion of the
universe.We present a complete renormalization scheme for the equation of
motion and the energy momentum tensor in flat cosmologies. We then consider an
O(N) inflaton model coupled self-consistently to gravity in the semiclassical
approximation, with `new inflation' type initial conditions. We study the
dynamics self-consistently and non-perturbatively with non-equilibrium field
theory methods in the large N limit. We find that spinodal instabilities drive
the growth of non-perturbatively large quantum fluctuations which shut off the
inflationary growth of the scale factor. A very specific combination of these
large quantum fluctuations plus the inflaton zero mode assemble into a new
effective field. This new field behaves classically and it is the object which
actually rolls down. The metric perturbations during inflation are computed
using this effective field and the Bardeen variable for superhorizon modes
during inflation. We compute the amplitude and index for the spectrum of scalar
density and tensorperturbations and find for these models that the spinodal
instabilities are responsible for a `red' primordial spectrum.
