  We establish a general slice theorem for the action of a locally convex Lie
group on a locally convex manifold, which generalizes the classical slice
theorem of Palais to infinite dimensions.
  We discuss two important settings under which the assumptions of this theorem
are fulfilled. First, using Gl\"ockner's inverse function theorem, we show that
the linear action of a compact Lie group on a Fr\'echet space admits a slice.
Second, using the Nash--Moser theorem, we establish a slice theorem for the
tame action of a tame Fr\'echet Lie group on a tame Fr\'echet manifold. For
this purpose, we develop the concept of a graded Riemannian metric, which
allows the construction of a path-length metric compatible with the manifold
topology and of a local addition.
  Finally, generalizing a classical result in finite dimensions, we prove that
the existence of a slice implies that the decomposition of the manifold into
orbit types of the group action is a stratification.
