  We consider the energy-supercritical nonlinear wave equation $u_{tt}-\Delta
u+|u|^2u=0$ with defocusing cubic nonlinearity in dimension $d=5$ with no
radial assumption on the initial data. We prove that a uniform-in-time {\it a
priori} bound on the critical norm implies that solutions exist globally in
time and scatter at infinity in both time directions. Together with our earlier
works in dimensions $d\geq 6$ with general data and dimension $d=5$ with radial
data, the present work completes the study of global well-posedness and
scattering in the energy-supercritical regime for the cubic nonlinearity under
the assumption of uniform-in-time control over the critical norm.
