In 1996 Rozansky and Witten described a new family of $(2+1)$-dimensional topological quantum field theories, quite different from the now familiar Chern-Simons theories. Instead of starting from a compact Lie group, one starts with a hyperk\"ahler manifold $X^{4n}$; the partition function (a topological invariant) for a closed $3$-manifold $M$ is then expressed as an integral over the space of all maps from $M$ to $X$. Further analysis shows that these invariants amount to evaluations of the universal finite-type invariant of Le, Murakami and Ohtsuki, using weight systems derived purely from the hyperk\"ahler manifold $X$.

I will explain the geometrical origin of these weight systems and then describe (joint work with Simon Willerton and Justin Sawon) a precise analogy between hyperk\"ahler manifolds and Lie algebras, the connections with Vassiliev theory, and the rigorous construction of the TQFT arising from $X$. The flavour of the theory is appealingly algebro-geometrical: whereas constructions of Chern-Simons theory start from the category of representations of a quantum group, Rozansky-Witten theory turns out to be based on the derived category of coherent sheaves on $X$.