This habilitation thesis is based on our work (D) M. Dirbák, First cohomology groups of minimal flows, Dissertationes Math. 562 (2021), 206 pp., which is also a part of the thesis as an attachment. Our interest in the thesis is in group extensions of minimal flows with compact abelian groups in the fibres. We study their structure from categorical and algebraic points of view, and describe relations of their dynamics to the one-dimensional algebraic-topological invariants. We determine the first cohomology groups of flows with simply connected acting groups and those of topologically free flows possessing a free cycle. As an application we show that minimal extensions of these flows not only do exist, but they have a rich algebraic structure.