Field-assisted electron emission has important technological applications, amongst them electron microscopy, electronic displays and space-vehicle neutralisers, but needs to be avoided in the context of electrical breakdown in vacuum. Ever since the publication of Fowler and Nordheim's 1928 paper on "Electron emission in intense electric fields", the problem of accurately modelling the field-assisted emission of electrons from a conductor has provided a rich field for investigation. Exact mathematical solution of the Schroedinger equation for the "transmission" and "tunnelling" processes involved is far from straightforward, because the resulting differential equation is non-linear; only in recent years have the "connection" problem and Stokes' Phenomenon begun to be sufficiently well-understood to allow definitive progress. Approximate emission theory has existed for many years. In order to establish a firm basis for accurate emission theory we are looking at the fundamental mathematical problems involved.