We have studied two mechanisms thought to be responsible for the anomalous currents and resulting charge accumulation on the surface of polymeric dielectrics. First, we find the detailed normal electric fields on the plane surface of an insulating dielectric next to an electrode by conformal and numerical techniques. The polarization, as determined by the calculated field, is found to be consistent with experimental surface charge profiles; they both fall off inversely as the distance from the electrode. However, polarization is shown to give only a small contribution to the observed effect. Second, a transmission line model of the surface resistivity is studied. Previous work on this model is shown to be misleading. We establish that the surface resistivity must vary with distance from the electrode in order to account for the observed inverse time dependence of the current at long times. This unphysical requirement, and some contradictory experimental data, indicate that the present understanding of surface charging is inadequate.