  The Vlasov--Maxwell equations are used for the kinetic description of
magnetized plasmas. As they are posed in an up to 3+3 dimensional phase space,
solving this problem is extremely expensive from a computational point of view.
In this paper, we exploit the low-rank structure in the solution of the Vlasov
equation. More specifically, we consider the Vlasov--Maxwell system and propose
a dynamic low-rank integrator. The key idea is to approximate the dynamics of
the system by constraining it to a low-rank manifold. This is accomplished by a
projection onto the tangent space. There, the dynamics is represented by the
low-rank factors, which are determined by solving lower-dimensional partial
differential equations. The proposed scheme performs well in numerical
experiments and succeeds in capturing the main features of the plasma dynamics.
We demonstrate this good behavior for a range of test problems. The coupling of
the Vlasov equation with the Maxwell system, however, introduces additional
challenges. In particular, the divergence of the electric field resulting from
Maxwell's equations is not consistent with the charge density computed from the
Vlasov equation. We propose a correction based on Lagrange multipliers which
enforces Gauss' law up to machine precision.
