Thesis (Ph.D.) - Indiana University, Department of Physics/University Graduate School, 2019Recent advances in theory and experiment have renewed efforts in the phenomenological under-standing of scattering for hadronic systems. Calculations in lattice QCD have shown that resonances can be extracted from the finite volume spectrum. These observables require analytic models to continue the amplitude to the complex energy plane to determine the infinite volume hadron spectrum. The forefront of this research is determining scattering observables for three particles from the finite volume spectra. We present an analytic representation for the elastic three-body scattering amplitude, which can be effectively used in lattice QCD calculations of amplitudes. Aspects of its analytic properties are discussed, specifically the one-particle exchange effects and triangle singularities, which are possible explanations for the many observed exotic hadrons.