  We introduce an equivariant version of the Cuntz semigroup, that takes an
action of a compact group into account. The equivariant Cuntz semigroup is
naturally a semimodule over the representation semiring of the given group.
Moreover, this semimodule satisfies a number of additional structural
properties. We show that the equivariant Cuntz semigroup, as a functor, is
continuous and stable. Moreover, cocycle conjugate actions have isomorphic
associated equivariant Cuntz semigroups. One of our main results is an analog
of Julg's theorem: the equivariant Cuntz semigroup is canonically isomorphic to
the Cuntz semigroup of the crossed product. We compute the induced semimodule
structure on the crossed product, which in the abelian case is given by the
dual action. As an application of our results, we show that freeness of a
compact Lie group action on a compact Hausdorff space can be characterized in
terms of a canonically defined map into the equivariant Cuntz semigroup,
extending results of Atiyah and Segal for equivariant $K$-theory. Finally, we
use the equivariant Cuntz semigroup to classify locally representable actions
on direct limits of one-dimensional NCCW-complexes, generalizing work of
Handelman and Rossmann.
