In the last two decades, the quark model lore has been challenged by unexpected exotic resonances found in experiments. Tetraquarks, pentaquarks, molecules, hybrids, and glueball candidates sprung forth and revitalized the field of hadron spectroscopy. Many of the most intriguing states decay into three or more particles. Determination of their masses and widths requires the study of related reaction amplitudes via analytic continuation to the complex energy plane. During the talk, I will review recent developments in the Lattice QCD formalisms of multi-particle scattering, focusing on the functional properties of the on-shell amplitudes. I will discuss the analytic continuation of the three-body scattering amplitude using an example of a scattering process where a bound state is formed in the two-body subchannel.