Lie algebras and the (co)homology of configuration spaces

We survey decades of research identifying the (co)homology of configuration spaces with Lie algebra (co)homology. The different routes to this one proto-theorem offer genuinely different explanations of its truth, and we attempt to convey some sense of the conceptual core of each perspective. We close with a list of problems. This article is part of the theme issue 'Derived Lie algebras in geometry and topology'.