We review the problem of plasma penetration of astrophysical magnetospheres, focusing on close binary stellar systems in which the Roche lobe overflow occurs. We point out that the accretion stream can penetrate the magnetosphere, even if its ram pressure is smaller than the magnetic pressure. This can occur without any turbulent mixing postulated in the past. The particular mechanism of cross-field penetration is an E × B drift caused by an electric field arising from polarization currents in the accretion stream. We determine the criteria for deep magnetosphere penetration; in particular, we find that deep penetration results for narrow streams (such as suggested by the previous theoretical analyses), and for reasonably low ambient plasma densities and temperatures.