In this paper pseudospectral methods are applied to the solution of electromagnetic problems. Specifically, the Chebyshev collocation method is used along with a mapping of the grid to relax the restrictions on the time-step. Entire-domain as well as multi-domain approaches are considered. The system of ordinary differential equations that comes from the spectral method formulation is solved using either the standard Runge-Kutta method of order four (RK4), or a newly developed algorithm from the class of diagonally implicit multistage integration methods. The accuracy of the spectral methods for all the different approaches is compared with that of the classical second-order FDTD scheme. Additionally, the CPU time as well as the memory required to achieve certain accuracy are reported for both the spectral and the FDTD methods.