Propagation of a directed pulse of electromagnetic radiation in a vacuum and in a rarefied, temporally dispersive medium is investigated. The pulse belongs to the class of broadband directed radiation beams termed electromagnetic ‘‘missiles’’ and is superposed on a carrier wave. First the propagation characteristics of the pulse in a vacuum are reviewed. In a vacuum the far‐field time‐integrated Poynting fluxU(z) of the pulse drops off as 1/z1+4&ngr;, with ‖&ngr;‖≤1/4. In the corresponding expression forU(z) in a dispersive medium the leading term also varies as 1/z1+4&ngr;since the high‐frequency end of the spectrum experiences very little dispersion. In a dispersive medium the propagation of the precursors and of the carrier signal is examined in certain limits and expressions for the spatial variation of the vector potential are given.