Physics shows that energy is necessary for economic production and, therefore, economic growth but the mainstream theory of economic growth, except for specialized resource economics models, pays no attention to the role of energy. This paper reviews the relevant biophysical theory and mainstream, resource economics, and ecological economics models of growth. A possible synthesis of energy-based and mainstream models is presented. This model shows that when energy is scarce it imposes a strong constraint on the growth of the economy but when energy is abundant its effect on economic growth is much reduced. This explains the industrial revolution as a releasing of the constraints on economic growth due to the development of methods of using coal and the discovery of new fossil fuel resources. Time series analysis shows that energy and GDP cointegrate and energy use Granger causes GDP when capital and other production inputs are included in the vector autoregression model. There are, however, various mechanisms that can weaken the links between energy and growth. The empirical literature finds that energy used per unit of economic output has declined in developed and some developing countries, due to both technological change and to a shift from poorer quality fuels such as coal to the use of higher quality fuels, and especially electricity. Substitution of other inputs for energy and sectoral shifts in economic activity play smaller roles.