  We prove global regularity, scattering and a priori bounds for the energy
critical Maxwell-Klein-Gordon equation relative to the Coulomb gauge on
(1+4)-dimensional Minkowski space. The proof is based upon a modified
Bahouri-Gerard profile decomposition [1] and a concentration
compactness/rigidity argument by Kenig-Merle [10], following the method
developed by the first author and Schlag [20] in the context of critical wave
maps.
