A result of Margulis says that every lattice in a simple Lie group G with rank\(_{{\mathbb{R}}}G>2\) is arithmetic. Up to local isomorphism it remains to consider the following non-compact groups (groups with rank\(_{{\mathbb{R}}}=1):\) \(O(n,1)\), \(U(n,1)\), and their quaternion and Cayley analogues. Non-arithmetic lattices in \(SU(2,1)\) and \(SU(3,1)\) were constructed by \textit{G. Mostov} using reflections in complex hyperplanes [cf. Elie Cartan et les mathématiques d'aujourd'hui, Astérisque, No.Hors. Sér. 1985, 289-309 (1985; Zbl 0605.22008)]. In the other case of the hyperbolic space examples of non-arithmetic lattices (for \(n=3,4,5)\) were found by Makarov, Nikulin and Vinberg. The paper under review provides a general construction of non-arithmetic lattices (cocompact and non-cocompact) in the projective orthogonal group \(PO(n,1)=O(n,1)/(\pm 1)\) for all \(n=2,3,...\). By taking two torsion free arithmetic subgroups of \(PO(n,1)\) and gluing together two submanifolds \(V^+_ i\) with boundary of dimension n of the corresponding hyperbolic manifolds \(V_ i\) along the (n-1)-dimensional boundary \(\partial V_ i^+\) (which is assumed to be totally geodesic in \(V_ i)\) by means of an isometry \(\partial V_ 1^+{\tilde \to}\partial V_ 2^+\) the authors produce a hybrid manifold V. The universal covering of V turns out to be the hyperbolic space and the fundamental group of V is a lattice in the isometry group PO(n,1) of the hyperbolic space. In the relevant cases the fundamental group \(\Gamma^+_ i\) of \(V^+_ i\) is Zariski dense in \(PO(n,1)\circ\). This implies the following commensurability property: If the group \(\Gamma\) is arithmetic then the groups \(\Gamma\) and \(\Gamma_ i\) are commensurable. In turn, one obtains a non-arithmetic lattice \(\Gamma\) by starting with two non-commensurable groups \(\Gamma_ 1\) and \(\Gamma_ 2\).