  In this report we introduce the basic techniques (of the Closed Time Path -
Coarse Grained Effective Action CTP-CGEA) and ideas (scaling, coarse-graining
and backreaction) behind the treatment of quantum processes in dynamical
background spacetimes and fields. We show how they are useful for the
construction of renormalization group (RG) theories for studying these
nonequilibrium processes and discuss the underlying issues. Examples are drawn
from quantum field processes in an inflationary universe, semiclassical
cosmology and stochastic gravity. In Part I we show how eternal inflation can
be treated as static critical phenomena, while a `slow-roll' or power-law
inflation can be treated as dynamical critical phenomena. In Part II we
introduce the key concepts in open systems and discuss the relation of
coarse-graining and backreaction. In Part III we present the CTP - CGEA. We
perform perturbative and nonperturbative evaluations, and show how to derive RG
equations. In Part IV, we use the RG equations to derive the Einstein-Langevin
equation in stochastic semiclassical gravity. We end with a discussion on why a
stochastic component of RG equations is expected for nonequilibrium processes
