The classification of a very robust class of simple C*-algebras---in the face of what might seem to be an inherent unlikelihood---would now appear to be complete.
The story has unfolded more or less in parallel with a similar story for von Neumann algebras---with a more or less similar specification of the class (separability, amenability, in the appropriate sense, and for the C*-algebras an additional, surprisingly painless, regularity requirement), and a more or less similar description of the classification functor.
Because of direct integral theory, the von Neumann algebra classification is not restricted to the simple case, but for the time being at least, the analogous classification of non-simple C*-algebras has proceeded in a somewhat uneven way, ironically in view of the early result of Gelfand and Naimark in the commutative case (the basis for all later non-simple cases).