I will introduce a notion of covering dimension for abstract Cuntz semigroups. When applied to the Cuntz semigroup of a C*-algebra, this dimension is always bounded by the nuclear dimension of the underlying algebra. If the C*-algebra is subhomogeneous, both dimensions agree.

The Cuntz semigroup of a separable, simple, $\mathcal{Z}$-stable C*-algebra is zero-dimensional if and only if the algebra has real rank zero or if it is stably projectionless. Further, I will also show that the Cuntz semigroups of $\mathcal{W}$-stable C*-algebras have dimension zero, and that $\mathcal{Z}$-stable C*-algebras have dimension at most one.