  In this work we prove global well-posedness for the massive Maxwell-Dirac
system in the Lorenz gauge in $\mathbb{R}^{1+3}$, for small, sufficiently
smooth and decaying initial data, as well as modified scattering for the
solutions. Heuristically we exploit the close connection between the massive
Maxwell-Dirac and the wave-Klein-Gordon equations, while developing a novel
approach which applies directly at the level of the Dirac equations. The
modified scattering result follows from a precise description of the asymptotic
behavior of the solutions inside the light cone, which we derive via the method
of testing with wave packets of Ifrim-Tataru.
