Stable moduli spaces of high-dimensional manifolds

Madsen and Weiss' theorem---identifying the stable homology of moduli
spaces of Riemann surfaces---and the theorem of Barratt--Priddy,
Quillen, and Segal---identifying the stable homology of classifying
spaces of symmetric groups---fit into a natural hierarchy of
statements concerning the homology of moduli spaces of (n-1)-connected
2n-manifolds. I will discuss recent joint work with Soren Galatius
which as a special case proves these statements for all n = 3, 4, ...