We present a survey of a very incomplete subject. The best-developed and simplest theories for current collection in magnetic fields are steady-state collisionless theories, and there must be understood before deparatures from them can be analyzed usefully, so we begin with a review of them. Recent numerical results indicate that steady-state collisionless Laplace-limit currents remain substantially below the Parker-Murphy (1967) canonical upper bound out to very large electrode potentials and approach it as a limit only very slowly if at all. Attempts to correct this theory for space-charge effects lead to potential disturbances which extend to infinite distance along the electrode's magnetic shadow, unless collisional effects are also taken into account. However, even a small amount of relative plasma drift motion, such as that involved in a typical rocket experiment, can change this conclusion fundamentally. It is widely believed that time-averaged current collection may be increased by effects of plasma turbulence, and we review the available evidence for and against this contention. Steady-state collisionless particle dynamics predicts the existence of a toroidal region of trapped orbits which surrounds the electrode. Light emissions from this region have been photographed, indicating that collisional ionization may also occur there, and this, and/or scattering by collisions or possibly turbulent fluctuations in this region, may also increase current collection by the electrode. We also discuss effects on particle motions near the electrode, assoicated with ''breakdown of magnetic insulation'' in the region of large electric fields near it. ¿American Geophysical Union 1992