CONTINUOUS flows of characteristic O+, O-, and 0’ were introduced by Ahmad [l]. In the same paper he also characterized planar continuous flows of characteristic 0’. Then Lam [7] gave a structure theorem for discrete and continuous flows which are of characteristic 0’ except for a zero-dimensional set. This paper is concerned with extending this result of Lam to a transformation group (X, 7’) where T is a generative group [4]. For this purpose we define a new notion of ‘strong characteristic 0’. One can also look at the results in this paper as a generalization of similar results for ‘nearly equicontinuous’ transformation groups [8], where equicontinuity is replaced by the weaker property of strong characteristic 0 but the phase group is restricted to a generative group. To lift the above mentioned properties from dynamical systems theory to topological dynamics one is usually faced with the problem of time orientation. For in the former the phase group is the additive group of reals which is endowed with a natural order relation, while in the latter this orientation of time is not available. In [2] the first author gave a scheme to lift concepts that have negative and positive versions from dynamical systems theory to topological dynamics. In particular the concept of characteristic O+ is generalized to that of P-characteristic 0, where P is a replete semigroup in the phase group. The concept of characteristic 0’ is generalized to that of 9-characteristic 0, that is, of P-characteristic 0 for all P E 9, where 9 is the set of all replete semigroups P in the phase group. In section 1 we study flows of 8-characteristic 0 and a decomposition theorem for such flows is given (theorem 1.6). Then we turn our attention to flows of almost characteristic 0, that is, flows of 8characteristic 0 except for a totally disconnected set. It is shown that under certain conditions the exceptional set in a discrete or continuous flow of almost characteristic 0 consists of at most two points (theorem 2.4 and remark 2.5). This improves the result in [7, section 171 since our theorems do not assume that the phase space is metric or that the exceptional set consists of recurrent points. The rest of the paper is devoted to studying a transformation group (X, T), where T is a generative group, and T is of strong Y-characteristic 0, except for a special set of points F.