We prove global well-posedness of the $(d+1)$-dimensional $(d\geq 4)$ massless Maxwell-Dirac equation in Coulomb gauge for data with small scale-critical Sobolev norm. A key step is to exploit (and justify) a deep analogy between Maxwell-Dirac and Maxwell-Klein-Gordon (for which an analogous result was proved earlier by Krieger-Sterbenz-Tataru), which says that the most difficult part of Maxwell-Dirac takes essentially the same form as Maxwell-Klein-Gordon. This is a joint work with C. Gavrus.