A minimal flow is a jointly continuous action of a topological group on a compact Hausdorff space such that every orbit is dense. We will present a survey of results on the structure of minimal flows, as well as steps towards a classification.

Certain classes, in particular those which are equicontinuous and distal, have been intensively studied, and we will discuss the structure of these flows. A "Galois theory", based on subgroups of the automorphism group of the universal minimal flow, provides a partial classification of minimal flows. More refined classification depends on a greater understanding of the proximal relation. To this end, we make use of the capturing operation, a kind of reverse orbit closure, which was introduced by Glasner and myself to characterize the distal and equicontinuous structure relations.