  The `Folk Theorem' that a smooth action by a compact Lie group can be
(canonically) resolved, by iterated blow up, to have unique isotropy type is
proved in the context of manifolds with corners. This procedure is shown to
capture the simultaneous resolution of all isotropy types in a `resolution
tower' which projects to a resolution, with iterated boundary fibration, of the
quotient. Equivariant K-theory and the Cartan model for equivariant cohomology
are tracked under the resolution procedure as is the delocalized equivariant
cohomology of Baum, Brylinski and MacPherson. This leads to resolved models for
each of these cohomology theories, in terms of relative objects over the
resolution tower and hence to reduced models as flat-twisted relative objects
over the resolution of the quotient. As a result an explicit equivariant Chern
character is constructed, essentially as in the non-equivariant case, over the
resolution of the quotient.
