  I present a proof of Kirchberg's classification theorem: two separable,
nuclear, $\mathcal O_\infty$-stable $C^\ast$-algebras are stably isomorphic if
and only if they are ideal-related $KK$-equivalent. In particular, this
provides a more elementary proof of the Kirchberg--Phillips theorem which is
isolated in the paper to increase readability of this important special case.
