  A general slice theorem for the action of a Fr\'echet Lie group on a
Fr\'echet manifolds is established. The Nash-Moser theorem provides the
fundamental tool to generalize the result of Palais to this
infinite-dimensional setting. The presented slice theorem is illustrated by its
application to gauge theories: the action of the gauge transformation group
admits smooth slices at every point and thus the gauge orbit space is
stratified by Fr\'echet manifolds.
  Furthermore, a covariant and symplectic formulation of classical field theory
is proposed and extensively discussed. At the root of this novel framework is
the incorporation of field degrees of freedom F and spacetime M into the
product manifold F * M. The induced bigrading of differential forms is used in
order to carry over the usual symplectic theory to this new setting. The
examples of the Klein-Gordon field and general Yang-Mills theory illustrate
that the presented approach conveniently handles the occurring symmetries.
