  We introduce and systematically study the notion of Rokhlin dimension (with
and without commuting towers) for compact group actions on $C^*$-algebras. This
notion generalizes the one introduced by Hirshberg, Winter and Zacharias for
finite groups, and contains the Rokhlin property as the zero dimensional case.
We show, by means of an example, that commuting towers cannot always be
arranged, even in the absence of $K$-theoretic obstructions. For a compact Lie
group action on a compact Hausdorff space, freeness is equivalent to finite
Rokhlin dimension of the induced action. We compare the notion of finite
Rokhlin dimension to other existing definitions of noncommutative freeness for
compact group actions. We obtain further $K$-theoretic obstructions to having
an action of a non-finite compact Lie group with finite Rokhlin dimension with
commuting towers, and use them to confirm a conjecture of Phillips.
