  In this Thesis we study quantum corrections to the classical dynamics for
mean values in field theory. To that end we make use of the formalism of the
closed time path effective action to get real and causal equations of motion.
  We introduce a coarse grained effective action, which is useful in the study
of phase transitions in field theory. We derive an exact renormalization group
equation that describes how this action varies with the coarse graining scale.
We develop different approximation methods to solve that equation, and we
obtain non perturbative improvements to the effective potential for a self
interacting scalar field theory. We also discuss the stochastic aspects
contained in this action.
  On the other hand, using the effective action, we find low energy and large
distance quantum corrections for the gravitational potential, treating
relativity as an effective low energy theory. We include the effect of scalar
fields, fermions and gravitons. The inclusion of metric fluctuations causes
Einstein semiclassical equations to depend on the gauge fixing parameters, and
they are therefore non physical. We solve this problem identifying as a
physical observable the trayectory of a test particle. We explicitly show that
the geodesic equation for such particle is independent of the arbitrary
parameters of the gauge fixing.
