  We present a quantization condition for the spectrum of a system composed of
three identical bosons in a finite volume with periodic boundary conditions.
This condition gives a relation between the finite volume spectrum and infinite
volume scattering amplitudes. The quantization condition presented is an
integral equation that in general must be solved numerically. However, for
systems with an attractive two-body force that supports a two-body bound-state,
a diboson, and for energies below the diboson breakup, the quantization
condition reduces to the well-known Luscher formula with exponential
corrections in volume that scale with the diboson binding momentum. To
accurately determine infinite volume phase shifts, it is necessary to
extrapolate the phase shifts obtained from the Luscher formula for the
boson-diboson system to the infinite volume limit. For energies above the
breakup threshold, or for systems with no two-body bound-state (with only
scattering states and resonances) the Luscher formula gets power-law volume
corrections and consequently fails to describe the three-particle system. These
corrections are nonperturbatively included in the quantization condition
presented.
