  We introduce the nuclear dimension of a C*-algebra; this is a noncommutative
version of topological covering dimension based on a modification of the
earlier concept of decomposition rank. Our notion behaves well with respect to
inductive limits, tensor products, hereditary subalgebras (hence ideals),
quotients, and even extensions. It can be computed for many examples; in
particular, it is finite for all UCT Kirchberg algebras. In fact, all classes
of nuclear C*-algebras which have so far been successfully classified consist
of examples with finite nuclear dimension, and it turns out that finite nuclear
dimension implies many properties relevant for the classification program.
Surprisingly, the concept is also linked to coarse geometry, since for a
discrete metric space of bounded geometry the nuclear dimension of the
associated uniform Roe algebra is dominated by the asymptotic dimension of the
underlying space.
