  Most strong-interaction resonances have decay channels involving three or
more particles, including many of the recently discovered $X$, $Y$ and $Z$
resonances. In order to study such resonances from first principles using
lattice QCD, one must understand finite-volume effects for three particles in
the cubic box used in calculations. Here we review efforts to develop a
three-particle quantization condition that relates finite-volume energies to
infinite-volume scattering amplitudes. We describe in detail the three
approaches that have been followed, and present new results on the relationship
between the corresponding results. We show examples of the numerical
implementation of all three approaches and point out the important issues that
remain to be resolved.
