  This is a combined write-up for two talks which were given consecutively and
which described different aspects of the same topic. We present a
generalization of L\"uscher's relation between the finite-volume spectrum and
S-matrix to three particles. Specifically, we consider a scalar field theory,
which has a $\mathbb{Z}_2$ symmetry that prevents even/odd coupling. The theory
is assumed to have no two-particle bound states and to have two-particle phase
shifts that are bounded by $\pi/2$ in the regime of elastic scattering.
Considering center of mass energies between one and five particle masses, we
evaluate a three-to-three finite-volume correlator to all orders in
perturbation theory. Only terms which are exponentially suppressed in volume
are neglected. From poles in the correlator we then determine the relation
between finite-volume spectrum and scattering quantities. In this analysis one
must carefully treat the unitary cusp at two-particle threshold. This point,
which was neglected in the conference talks, is described in some detail here.
We also describe an important check on our main result by reproducing the large
volume expansion of the energy shift from the three-particle threshold. This is
found to be consistent with previous work through four non-trivial orders.
