Characteristic classes of manifold bundles are an important tool to distinguish families of manifolds, much like characteristic classes of vector bundles. In the context of even-dimensional manifolds, the ring of characteristic classes of manifold bundles which are stable, under a certain notion of stabilization, was completely described by Galatius and Randal-Williams in terms of purely homotopy theoretic invariants. This provided a higher dimensional generalization of Madsen and Weiss' resolution of the Mumford conjecture from algebraic geometry. I will describe an odd-dimensional analog of Galatius and Randal-Williams' result for an appropriate notion of stabilization.