In 1974 Kadison and Kastler introduced a notion of closeness of operator algebras in terms of the Hausdorff distance between their unit balls. They raised the question of whether two close algebras are isomorphic, or even unitarily equvalent. In the context of von Neumann algebras, we say that M is KK-stable if it is isomorphic to any suitably close algebra. KK-stability is known for all amenable von Neumann algebras, a result due to Christensen. In this talk I will describe the first known classes of nonamenable KK-stable von Neumann algebras. These arise as crossed products by certain nonamenable groups. The talk will be aimed at a general audience. This is joint work with Jan Cameron, Erik Christensen, Allan Sinclair, Stuart White and Alan Wiggins.