  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.
