Hydrodynamic or continuum descriptions of electron transport have long been used for modeling and simulating semiconductor devices. In this paper, we use classical field theory ideas to discuss the physical foundations of such descriptions as applied specifically to high-field transport regimes. The classical field theory development of these types of models is of interest because it differs significantly from and may be viewed as complementary to conventional derivations based on the Boltzmann equation: After outlining the general field theoretic principles upon which our development of fluid-based high-field transport descriptions is based, we study several specific models both analytically and using numerical simulation. These models provide an overall framework for understanding and extending various theories which have appeared in the literature. Most importantly, they emphasize the importance of including memory or history effects and viscosity in describing high-field transport. In all of this our aim is a unified and synoptic view unencumbered with microscopic details. Obtaining quantitative agreement with specific experiments and/or microscopic simulations is only of secondary importance. We share the view that continuum approaches can provide succinct and computationally-efficient models needed for current and future semiconductor device analysis and engineering. At the same time, we believe that these models need not be phenomenological but can be given solid physical foundation in macroscopic principles.