  We prove an analogue of the Madsen-Weiss theorem for high dimensional
manifolds. For example, we explicitly describe the ring of characteristic
classes of smooth fibre bundles whose fibres are connected sums of g copies of
S^n x S^n, in the limit as g tends to infinity. Rationally it is a polynomial
ring in certain explicit generators, giving a high dimensional analogue of
Mumford's conjecture.
  More generally, we study a moduli space N(P) of those null-bordisms of a
fixed (2n-1)-dimensional manifold P which are highly connected relative to P.
We determine the homology of N(P) after stabilisation using certain
self-bordisms of P. The stable homology is identified with that of a certain
infinite loop space.
