We review past research on the propagation of electromagnetic pulse signals over the surface of the earth with emphasis on analytical methods to predict waveforms. The best example is the electromagnetic field radiated from a lightning stroke that can be observed at global distances or very close in if one has sufficient motivation. In any case, the propagation channel will distort the signal wave shape because of the strong frequency dependence of the attenuation factors and the phase velocities of the modes. We begin with a flat earth model that has the virtue of simplicity particularly when displacement currents in the earth are relatively small compared with the conduction currents. Here we see that the rise time of the ideally radiated step function signal is of the order of a few microseconds at a range of 50 km over average land paths. Such a prediction is in accord with published experimental data. At greater ranges, earth curvature comes into play and we show here how various analytical and numerical methods can be used in overlapping time regions which serves as a good consistency check. Other complicated effects such as mixed land-sea paths and ionospheric influences are also reviewed.