In this talk, we will outline a proof that solutions to the defocusing cubic Klein-- Gordon equation $$ u_{tt} - \Delta u + u + u^3 = 0, $$ with real-valued initial data $u(0) \in H^1(\R^2)$ and $u_t(0) \in L^2(\R^2)$, scatter both forward and backward in time. We will also discuss a related result in the focusing case. Along the way, we will explain what we mean by `scattering' and will provide some motivation for the problem. This is joint work with Rowan Killip and Monica Vi\c{s}an.