The incompressible Navier–Stokes–Maxwell system is a classical model describing the evolution of a plasma (i.e. an electrically conducting fluid). Although small smooth solutions to this system (in the spirit of Fujita–Kato) are known to exist, the existence of large weak solutions (in the spirit of Leray) in the energy space remains unknown. This defect can be attributed to the difficulty of coupling the Navier–Stokes equations with a hyperbolic system. In this talk, we will describe recent results aiming at building solutions to Navier–Stokes–Maxwell systems in large functional spaces. In particular, we will show, for any initial data with finite energy, how a smallness condition on the electromagnetic field alone is sufficient to grant the existence of global solutions.