I will talk about Baut(X) from the point of view of rational homotopy theory, focusing in particular on simply connected manifolds X. In recent joint work with Ib Madsen we established homological stability results for Baut(X) for particular classes of such manifolds, and we showed that the stable cohomology may be calculated in terms of graph complexes à la Kontsevich. I will discuss these results and outline a general approach to the rational cohomology of Baut(X) informed by our findings. If time admits I will also discuss how our results shed new light on the classical approach to measuring the difference between Baut(X) and BDiff(X) through surgery theory and Waldhausen's A(X).