In this talk I will present the main results of a joint work with James Gabe: Given a countable discrete group G, two amenable and outer G-actions on stable Kirchberg algebras are cocycle conjugate precisely when they are equivariantly KK-equivalent. In fact we have a suitable version of such a theorem, albeit with more delicate assumptions, for G being any second-countable locally compact group. After discussing the core concepts playing a role in our theory, I will state the classification results in their most general form and mention a number of interesting corollaries. If time permits, I will try to illustrate some of the key ideas in very special cases.