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The demand for computing is continuing to grow exponentially. This growth will translate to exponential growth in computing's energy consumption unless improvements in its energy-efficiency can outpace increases in its demand. Yet, after decades of research, further improving energy-efficiency is becoming increasingly challenging, as it is already highly optimized. As a result, at some point, increases in computing demand are likely to outpace increases in its energy-efficiency, potentially by a wide margin. Such exponential growth, if left unchecked, will position computing as a substantial contributor to global carbon emissions. While prominent technology companies have recognized the problem and sought to reduce their carbon emissions, they understandably focus on their successes, which has the potential to inadvertently convey the false impression that this is now, or will soon be, a solved problem. Such false impressions can be counterproductive if they serve to discourage further research in this area, since, as we discuss, eliminating computing's, and more generally society's, carbon emissions is far from a solved problem. To better understand the problem's scope, this paper distills the fundamental trends that determine computing's carbon footprint and their implications for achieving sustainable computing.