The recent trend in New Economics is the establishment of measures of sustainable wealth and welfare which take into account all the parameters of economic, environmental, and social life and progress, juxtaposed to the conventional and myopic GDP. This review summarizes results from a series of recent papers in the energy-growth nexus field, which have perused a proxy for the sustainable GDP instead of the conventional GDP and discusses the difference in results and policy implications. The energy-growth nexus field itself has generated a bulk of work since the seminal study of Kraft and Kraft (1978), but still the field needs new perspectives in order to generate results with a consensus. The bidirectional causality between energy consumption and sustainable economy provides evidence for the Feedback Hypothesis, a statement that essentially warns that it is too early for sustainability to be feasible without fossil energy consumption, and vice versa. The unidirectional causality reveals, on the one side, that an economy cannot grow without the plentiful consumption of energy (the Growth Hypothesis) and, on the other side, that the growth of the economy fuels energy consumption (the Conservation Hypothesis). Failure to corroborate causality between energy consumption and economic growth is evidence for the Neutrality Hypothesis.