Some years ago F. Gehring and G. J. Martin introduced the notion of a convergence group acting on the 2-sphere. These are defined topologically to have the properties characteristic of Kleinian groups. The purpose of this paper is to classify those convergence groups which are most closely analogous to quasi-Fuchsian groups. The assumption made by the authors is that the ordinary set has two components whose union is invariant. (A more complicated condition is needed if one wants to capture the notion of `groups of the second kind'.) There may a priori be other components than these two. However under these assumptions the authors succeed in classifying the groups which arise and thereby extend a large body of earlier work by Kra, Maskit, Marden, Thurston and others. For the details of the results, which are quite delicate, we have to refer to the paper.