  We show that the Dixmier-Douady theory of continuous field $C^*$-algebras
with compact operators $\mathbb{K}$ as fibers extends significantly to a more
general theory of fields with fibers $A\otimes \mathbb{K}$ where $A$ is a
strongly self-absorbing C*-algebra. The classification of the corresponding
locally trivial fields involves a generalized cohomology theory which is
computable via the Atiyah-Hirzebruch spectral sequence. An important feature of
the general theory is the appearance of characteristic classes in higher
dimensions. We also give a necessary and sufficient $K$-theoretical condition
for local triviality of these continuous fields over spaces of finite covering
dimension.
