  We prove the existence of global solutions to the energy-supercritical wave
equation in R^{3+1} u_{tt}-\Delta u + |u|^N u = 0, u(0) = u_0, u_t(0) = u_1,
4<N<\infty, for a large class of radially symmetric finite-energy initial data.
Functions in this class are characterized as being outgoing under the linear
flow --- for a specific meaning of "outgoing" defined below. In particular, we
construct global solutions for initial data with large (even infinite) critical
Sobolev, Besov, Lebesgue, and Lorentz norms and several other large critical
norms.
