In 1970, Arnold proved that the homology groups of the braid groups on n strands stabilizes as n tends to infinity, a phenomenon called "homological stability". The pure braid groups, in contrast, are not homologically stable. In this (partly expository) talk I will describe a sense in which (co)homology groups of the pure braid groups do stabilize when we take into account the natural symmetric group actions. We will use tools from "representation stability" to shed light on the structure of the (co)homology of the pure braid groups, and many of their generalizations. This talk will survey work of Church, Ellenberg, and Farb, and joint work with Miller.