In this paper, we are concerned with the construction of global-in-time solutions of the Cauchy problem of the Vlasov-Maxwell-Boltzmann system near Maxwellians with strong uniform background magnetic field. The background magnetic field under our consideration can be any given non-zero constant vector rather than vacuum in the previous results available up to now. Our analysis is motivated by the nonlinear energy method developed recently in \cite{Guo-IUMJ-2004, Liu-Yang-Yu-Physica D-2004, Liu-Yu-CMP-2004} for the Boltzmann equation and the key point in our analysis is to deduce the dissipation estimates of the electronic field, especially the $L^2(\mathbb{R}^+\times\mathbb{R}^3)-$integrability of the electronic field itself. Such a difficulty is overcome by utilizing the dissipative structure of the macroscopic equations and by employing the negative Sobolev strategy introduced in \cite{Guo-Wang-CPDE-2012}. This is a joint work with Prof. H.-J. Zhao.