For a manifold P I will discuss the "moduli space of highly connected null bordisms of P". A null bordism of P is a compact manifold whose boundary is P, and the moduli space parametrizes such null bordisms. In joint work with Oscar Randal-Williams, we determine the stable cohomology of this moduli space when P is a manifold of odd dimension at least 5. Our answer is analogous to the Madsen-Weiss theorem, which is the case where P is a circle.