The article states some results obtained by the author in his thesis: The author considers compact 4-manifolds that are the quotient of some simply connected solvable Lie group \(S\) by an isometric action of a group \(\Gamma\). When \(S\) is solvable it has been shown that the isomorphism class of the fundamental group of such a manifold determines the diffeomorphism class in all dimensions with the possible exception of dimension four [\textit{F. T. Farrell} and \textit{L. E. Jones}, in `Geometry from the Pacific Rim', Proc. Conf. Singapore 1994, pp. 85-97 (1997; Zbl 0895.53047)]. In Chapter 2 of the thesis this result is extended to include all 4-dimensional compact infra-solvmanifolds with the possible exception of one anomalous class. The fundamental groups of those compact infra-solvmanifolds with a 4-dimensional model have been essentially classified with the exception of two models (one of which belongs to the infinite family). In Chapter 3 of the thesis the first effort towards the classification of the fundamental groups of infra-solvmanifolds admitting one of the remaining two models is made. Chapter 4 of the thesis completes the work initiated by \textit{M. Ue} [J. Math. Soc. Japan 42, No. 3, 511-540 (1990; Zbl 0707.57010)]: in classifying those total spaces of surface-bundles over surfaces with flat base and fibre (up to bundle-diffeomorphism) and thence showing that each such manifold admits an infra-solvmanifold geometry.