Let G be a group of Mobius transformations and V the space of com- plex polynomials of degree < some fixed even integer. Using the action of G on V defined by Eichler, we compute the dimension of the cohomology space H'(G, V), first for G an arbitrary F-group (a generalization of Fuchsian group) and then for the free product of finitely many F-groups. These results extend those which Eichler obtained in a 1957 paper, where a correspondence was established between elements of H'(G, V) and cusp forms on G. Introduction. In view of recent results in the theory of discontinuous groups, the cohomology of such groups has become an object of study in Riemann surface theory. For a Fuchsian group G with the usual presentation, Eichler (4) computed the dimension of a certain subspace of H'(G, V), V being a space of complex polynomials, and obtained a correspondence between cohomology classes and cusp forms on G. (The action of G on V is described in ?3.) The subspace consists of those cohomology classes represented by cocycles which are trivial on certain generators of G. More recently Bers (1) obtained further results in this direction for Kleinian groups, and it became a matter of interest to know the dimension of the full cohomology space. In this paper we obtain formulas for the dimension of H'(G, V) in terms of the parameters occurring in the presentation of G. In ?3 and ?4, this is carried out for F-groups (these include Fuchsian groups; see ?2) and in ?5 is extended to the free