Let \(\Gamma\) be a finitely generated non-elementary Kleinian group with region of discontinuity \(\Omega\) and limit set \(\Lambda\). Let \(A_ q(\Omega,\Gamma)\) be the Bers space of cusp forms of weight (-2q). Define the set \(\Lambda_ q\) to be the limit set \(\Lambda\) together with those fixed points z in \(\Omega\) of elliptic elements such that the stabilizer of z in \(\Gamma\) is a subgroup of order v, where v is not a factor of q- 1. Given this \(\Lambda_ q\) let \(R_ q(\Lambda_ q)\) be the space of rational functions f which are holomorphic on \(({\mathbb{C}}\cup \{\infty \})- \Lambda_ q\), all of the poles of f at finite points of \(\Lambda_ q\) are simple and \(f(z)=O(| z|^{-p})\) as \(z\to \infty\) where \(p=2q-1\) if \(\infty \in \Lambda_ q\) and \(p=2q\) otherwise. For f an element in \(R_ q(\Lambda_ q)\) it is well known that the Poincaré series \(\theta_ q(f,\Gamma))\) defined as \(\sum_{\gamma \in \Gamma}f(\gamma z)\gamma ^{\prime q}(z)\) converges absolutely and uniformly on compact subsets of \(\Omega\) and moreover \(\theta_ q(f,\Gamma)\) belongs to \(A_ q(\Omega,\Gamma).\) If S is a subset of \(\Lambda_ q\) then \(R_ q(S)\) denotes the set of f in \(R_ q(\Lambda_ q)\) which are holomorphic on \(\Lambda_ q-S\). \textit{L. Bers} [Commun. Pure. Appl. Math. 26, 667-672 (1973; Zbl 0277.30016)] has shown that \(\theta_ q:R_ q(\Lambda)\to A_ q(\Omega,\Gamma)\) is surjective. The author begins by developing a finite set S such that \(\theta_ q(R_ q(S))=A_ q(\Omega,\Gamma).\) The set S consists of \(\{\gamma_ J(a_ k):\quad 1\leq k\leq 2q-1,\quad 1\leq J\leq N\}\) where \(\{a_ k\}\) is a set of distinct points in \(\Lambda_ q\) and \(\gamma_ 0=id\), and \(\{\gamma_ J:\quad 0\leq J\leq N\}\) is a generating set for \(\Gamma\). This means that in many instances one can choose from a presentation of \(\Gamma\) of a set S counting of precisely K points, where \(K=2q-1+\dim A_ q(\Omega,\Gamma),\) such that \(\theta_ q:R_ q(S)\to A_ q(\Omega,\Gamma)\) is an isomorphism. The author then turns his attention to the much more interesting (and more difficult) Poincaré vanishing problem. This is stated as follows: Let \(\Delta\) be a \(\Gamma\) invariant union of components of \(\Omega\) and the problem is to find necessary and sufficient conditions so that \(\theta_ q(f,\Gamma)|_{\Delta}\) vanishes identically for a given f in \(R_ q(\Lambda_ q)\). He first shows that the solution to the problem can be reduced to describing a basis for a particular subspace of \(R_ q(\Lambda_ q)\). Now comes the major advance done in this work. By using the Eichler cohomology theory for Kleinian groups and in particular the Bers map as developed by Ahlfors, Bers and the author [see the author, Automorphic forms and Kleinian groups (1972; Zbl 0253.30015) for an excellent exposition of this theory], he is able to show that this basis can be explicitely determined algebraically from the parabolic cocycles for \(\Gamma\) whenever the Bers map is subjective. Thus for groups with simple algebraic presentation and for rational functions which have poles only at the loxodromic fixed points, the Poincaré vanishing problem has a purely algebraic solution. Since the Bers map is surjective whenever \(\Gamma\) is a Fuchsian, quasi-Fuchsian or a Schottky group one can develop a finite algebraic algorithm to decide if \(\theta_ q(f,\Gamma)\equiv 0\) for f in \(R_ q(\Lambda_ q)\). This paper is a major advance in the theory of automorphic forms and Kleinian groups. The author has done a beautiful job of presenting the paper in such a way as to make it extremely readable. This short review does not due full justice to this work.