  The existence of run-away solutions in classical and non-relativistic quantum
electrodynamics is reviewed. It is shown that the less singular high energy
behavior of relativistic spin 1/2 quantum electrodynamics precludes an
analogous behavior in that theory. However, a Landau-like anomalous pole in the
photon propagation function or in the electron-massive photon foward scattering
amplitude would generate a new run-away, characterized by an energy scale omega
~ m_e exp (1/alpha). This contrasts with the energy scale omega ~ (m_e/alpha)
associated with the classical and non-relativistic quantum run-aways.
