In this article, we examine the Fowler-Nordheim (FN) equation for field emission using pedagogical models to introduce and illuminate its origins, limitations, extensions, and application to multidimensional structures. The analyses of modern electron sources generally invoke either the FN equation or the Richardson-Laue-Dushman equation (thermionic emission) to interpret experimental data. These equations have ranges of validity that are increasingly challenged by operating conditions. The present article shall therefore have several aims. An introduction to and review of the FN equation shall be presented. Extensions to account for many body and other effects, shall be motivated by accessible models, and a generalized thermal-field emission methodology developed to account for low work function, high fields, photoexcitation, and other conditions in which the incident electron energy is near the barrier maximum. An account of effects such as resonance, which are not generally part of the standard emission lexicon, is given. Finally, specialized topics using the aforementioned analyses shall be examined, e.g., multidimensionality, the statistical nature of emission site variation, and so on. The analyses shall be predicated on simple models in an effort to provide formulae of general utility such that computational requirements are minimized.