  The mid-seventies' works on C*-algebras of Brown-Douglas-Fillmore and Elliott
both contained uniqueness and existence results in a now standard sense. These
papers served as keystones for two separate theories -- KK-theory and the
classification program -- which for many years parted ways with only moderate
interaction. But recent years have seen a fruitful interaction which has been
one of the main engines behind rapid progress in the classification program.
  In the present paper we take this interaction even further. We prove general
existence and uniqueness results using KK-theory and a concept of
quasidiagonality for representations. These results are employed to obtain new
classification results for certain classes of quasidiagonal C*-algebras
introduced by H. Lin. An important novel feature of these classes is that they
are defined by a certain local approximation property, rather than by an
inductive limit construction.
  Our existence and uniqueness results are in the spirit of classical
Ext-theory. The main complication overcome in the paper is to control the
stabilization which is necessary when one works with finite C*-algebras. In the
infinite case, where programs of this type have already been successfully
carried out, stabilization is unnecessary. Yet, our methods are sufficiently
versatile to allow us to reprove, from a handful of basic results, the
classification of purely infinite nuclear C*-algebras of Kirchberg and
Phillips.
  Indeed, it is our hope that this can be the starting point of a unified
approach to classification of nuclear C*-algebras.
