It is well-known that the signature of a manifold fibering over a surface is divisible by 4. One may ask if restricting the diffeomorphism type of the fiber M results in stronger divisibility constraints. I will present a synopsis of joint work with M. Krannich on the classification of total spaces of such fiber bundles if M is highly connected and almost parallelizable, including a calculation of their characteristic numbers. As a corollary, we determine the occurring signatures; it turns out that signature 4 is viable only in dimensions 4, 8, and 16. I will end by describing a basis of the second integral cohomology of BDiff(M) up to torsion in terms of generalized Miller--Morita--Mumford classes.