Abstract : This effort addressed the direct problem in electromagnetic wave propagation and scattering in dispersive dielectrics objects. Particularly, the research concentrated on the development of numerical approaches that can solve the problem in a timely fashion as accurately as possible given a reasonable amount of computational resources. In addition, analytical methods were developed for the purpose of extracting features of the problem which are special to dispersive objects. First, four existing Finite Difference Time Domain extensions for the modeling of pulse propagation in Debye or Lorentz dispersive media were analyzed through studying the stability and phase error properties of the coupled difference equations corresponding to Maxwell's equations and to the equations for the dispersion. In order to understand more fully the discretization requirements and the general behavior of existing numerical methods for dispersive media we considered the propagation of arbitrary electromagnetic pulses in anomalously dispersive dielectrics characterized by M relaxation processes. Armed with the knowledge that finite difference schemes, which are more accurate in space than in time, are better suited for modeling pulse propagation in dispersive dielectrics we next determined some important properties of such high order schemes in comparison to standard schemes. Finally, we examined further aspects of electromagnetic pulse propagation in anomalously dispersive media, using the Debye model as an example.