This paper reviews forty years of research applying econometric models of discrete-continuous choice to analyze residential demand for energy. The review is primarily from the perspective of economic theory. We examine how well the utility-theoretic models developed in the literature match data that is commonly available on residential energy use, and we highlight the modeling challenges that have arisen through efforts to match theory with data. The literature contains two different formalizations of a corner solution. The first, by Dubin and McFadden (1984) and Hanemann (1984), models an extreme corner solution, in which only one of the discrete choice alternatives is chosen. While those papers share similarities, they also have some differences which have not been noticed or exposited in the literature. Subsequently, a formulation first implemented by Wales and Woodland (1983) and extended by Kim et al. (2002) and Bhat (2008) models a general corner solution, where several but not all of the discrete choice alternatives are chosen. Seventeen papers have employed one or another of these models to analyze residential demand for fuels and/or energy end uses in a variety of countries. We review issues that arose in these applications and identify some alternative model formulations that can be used in future work on residential energy demand.