We introduce the completely positive rank, a notion of covering dimension for nuclear C ∗ -algebras and analyze some of its properties. The completely positive rank behaves nicely with respect to direct sums, quotients, ideals and inductive limits. For abelian C ∗ -algebras it coincides with covering dimension of the spectrum and there are similar results for continuous trace algebras. As it turns out, a C ∗ -algebra is zero-dimensional precisely if it is AF. We consider various examples, particularly of one-dimensional C ∗ -algebras, like the irrational rotation algebras, the Bunce–Deddens algebras or Blackadar's simple unital projectionless C ∗ -algebra. Finally, we compare the completely positive rank to other concepts of noncommutative covering dimension, such as stable or real rank.