  In this paper, error analysis is established for Runge-Kutta discontinuous
Galerkin (RKDG) methods to solve the Vlasov-Maxwell system. This nonlinear
hyperbolic system describes the time evolution of collisionless plasma
particles of a single species under the self-consistent electromagnetic field,
and it models many phenomena in both laboratory and astrophysical plasmas. The
methods involve a third order TVD Runge-Kutta discretization in time and upwind
discontinuous Galerkin discretizations of arbitrary order in phase domain. With
the assumption that the exact solution has sufficient regularity, the $L^2$
errors of the particle number density function as well as electric and magnetic
fields at any given time $T$ are bounded by $C h^{k+\frac{1}{2}}+C\tau^3$ under
a CFL condition $\tau /h \leq \gamma$. Here $k$ is the polynomial degree used
in phase space discretization, satisfying $k \geq \left \lceil \frac{d_x +
1}{2} \right \rceil$ (the smallest integer greater than or equal to
$\frac{d_x+1}{2}$, with $d_x$ being the dimension of spatial domain), $\tau$ is
the time step, and $h$ is the maximum mesh size in phase space. Both $C$ and
$\gamma$ are positive constants independent of $h$ and $\tau$, and they may
depend on the polynomial degree $k$, time $T$, the size of the phase domain,
certain mesh parameters, and some Sobolev norms of the exact solution. The
analysis can be extended to RKDG methods with other numerical fluxes and to
RKDG methods solving relativistic Vlasov-Maxwell equations.
