A group $G$ is isomorphic to the fundamental groups of a compact solvmanifold if and only if it fits into the short exact sequence $1\to N\to G\to\mathbb Z^n\to 1$ where $N$ is a finitely generated torsion-free nilpotent group. This is stated on p.253 and explained is chapter III of Auslander's An exposition of the structure of solvmanifolds. Part I: Algebraic theory. In particular, every torus bundle over a torus is a solvmanifold.

You may also be interested in Theorem 3 of Wilking's paper Rigidity of group actions on solvable Lie groups which gives an analogous result for infrasolvmanifolds.

A group $G$ is isomorphic to the fundamental groups of a compact solvmanifold if and only if it fits into the short exact sequence $1\to N\to G\to\mathbb Z^n\to 1$ where $N$ is a finitely generated torsion-free nilpotent group. This is stated on p.253 and explained is chapter III of Auslander's An exposition of the structure of solvmanifolds. Part I: Algebraic theory. In particular, every torus bundle over a torus is homotopy equivalent to a solvmanifold.

You may also be interested in Theorem 3 of Wilking's paper Rigidity of group actions on solvable Lie groups which gives an analogous result for infrasolvmanifolds.