Surface bundles are an excellent “proving ground” for topologists. They are complicated enough to be interesting, but also tangible enough to yield fruitful results. The structure of these manifolds is deeply connected to the mapping class groups of surfaces via the monodromy representation. We will share some of the history of this relationship and then present some of the progress on understanding how the curvature of surface bundles is informed by the geometry of subgroups of mapping class groups.