  We study smooth bundles over surfaces with highly connected almost
parallelizable fiber $M$ of even dimension, providing necessary conditions for
a manifold to be bordant to the total space of such a bundle and showing that,
in most cases, these conditions are also sufficient. Using this, we determine
the characteristic numbers realized by total spaces of bundles of this type,
deduce divisibility constraints on their signatures and $\hat{A}$-genera, and
compute the second integral cohomology of ${\rm BDiff}^+(M)$ up to torsion in
terms of generalized Miller--Morita--Mumford classes. We also prove analogous
results for topological bundles over surfaces with fiber $M$ and discuss the
resulting obstructions to smoothing them.
