We survey briefly the definition of the Rozansky–Witten invariants, and review their
relevance to the study of compact hyperkähler manifolds. We consider how various
generalisations of the invariants might prove useful for the study of non-compact
hyperkähler manifolds, of quaternionic Kähler manifolds, and of relations between
hyperkähler manifolds and Lie algebras. The paper concludes with a list of
additional problems.