  We prove a unique continuation from infinity theorem for regular waves of the
form $[ \Box + \mathcal{V} (t, x) ]\phi=0$. Under the assumption of no incoming
and no outgoing radiation on specific halves of past and future null
infinities, we show that the solution must vanish everywhere. The "no
radiation" assumption is captured in a specific, finite rate of decay which in
general depends on the $L^\infty$-profile of the potential $\mathcal{V}$. We
show that the result is optimal in many regards. These results are then
extended to certain power-law type nonlinear wave equations, where the order of
decay one must assume is independent of the size of the nonlinear term. These
results are obtained using a new family of global Carleman-type estimates on
the exterior of a null cone. A companion paper to this one explores further
applications of these new estimates to such nonlinear waves.
