In the search for new resonances and determination of properties of the known ones, three-body final states play an important role, for example in the axial mesons, exotic mesons, or the Roper resonance. An amplitude is presented that is manifestly three-body unitary and two-body unitary for the sub-amplitudes. Close connections to triangle singularities and Khuri-Treiman equations will be discussed. In the second part, the implementation in the finite volume will be presented that will allow for the infinite volume extrapolation of results of lattice QCD calculations on three-body systems.