We study a number of regularity properties of C*-algebras which are
intimately related in the case of nuclear C*-algebras.
These properties can be expressed topologically (as dimension type
conditions), C*-algebraically (as stability under tensoring with suitable
strongly self-absorbing C*-algebras), and at the level of homological
invariants (in terms of comparison properties of projections, or positive
elements, respectively).
We explain these concepts and some known relations between them,
and outline their relevance for the classification program. (As a particularly
satisfying application, one obtains a classification result for C*-algebras
associated to compact, finite-dimensional, minimal, uniquely ergodic,
dynamical systems.)
Furthermore, we investigate potential applications of these technologies
to other areas, such as coarse geometry.