I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here. Let $$ T^n \to M \to T^m $$ be a principal torus bundle over a torus. Then $ M $ is a solvmanifold, even a nilmanifold (in fact $ M $ is the total space of a principal torus bundle over a torus if and only if it is a compact nilmanifold for a 2 step nilpotent Lie group, this is theorem 3 of [Palais, Stewart, TORUS BUNDLES OVER A TORUS]). What if the bundle is not necessarily principal? Is every torus bundle over a torus a solvmanifold? In other words, if we have a fiber bundle $$ T^n \to M \to T^m $$ then can we conclude that the total space $ M $ is a solvmanifold? EDIT: The answer, comments and references from Igor Beledrek prove that something much stronger is true: A manifold $ M $ is the total space of a bundle $$ N \to M \to T^n $$ where $ N $ is a compact nilmanifold and $ T^n $ is a torus if and only if $ M $ is homeomorphic to a compact solvmanifold. The smooth case is also addressed. The answer, comments and references from Igor Beledrek demonstrate that already in dimension 4 there exists a smooth manifold $ M $ (in particular an exotic 4 torus) $$ T^2 \to M \to T^2 $$ which arise as the total space of a torus bundle over a torus (both base and fiber with standard smooth structure) but which is not diffeomorphic to any solvmanifold. So in every dimension $ d \geq 4 $ there are smooth torus bundles over a torus that are not diffeomorphic to any solvmanifold. For $ d \leq 3 $ we have by Moise's theorem that there are no exotic smooth structures. So all torus bundles over a torus with total space of dimension $ d \leq 3 $ are diffeomorphic to a solvmanifold.