A brief summary of the methods of solving transient electromagnetic wave problems in inhomogeneous media will be given. The two distinct general techniques, which are the inverse Fourier transformation of time-harmonic solutions and the direct time-domain formulation, will be illustrated by way of two examples. In the first example, an efficient numerical mode-matching method to obtain the response of an electromagnetic source in a two-dimensional cylindrical inhomogeneity is described. Using this method, the solution is first obtained in the frequency domain. The time-domain solution is then found by inverse Fourier transforming the frequency-domain solution. In the second method, a finite-difference scheme is used to find the transient response of a point source in a two-dimensional inhomogeneity. Two different methods are proposed to treat the source region singularity. Transmitting boundary conditions are applied on the walls of the finite difference grid so that a finite-sized box can be used to model an infinite region.