The notion is pervasive in the climate science community and in the public at large that the climate impacts of fossil fuel CO2 release will only persist for a few centuries. This conclusion has no basis in theory or models of the atmosphere / ocean carbon cycle, which we review here. The largest fraction of the CO2 recovery will take place on time scales of centuries, as CO2 invades the ocean, but a significant fraction of the fossil fuel CO2, ranging in published models in the literature from 20-60%, remains airborne for a thousand years or longer. Ultimate recovery takes place on time scales of hundreds of thousands of years, a geologic longevity typically associated in public perceptions with nuclear waste. The glacial / interglacial climate cycles demonstrate that ice sheets and sea level respond dramatically to millennial-timescale changes in climate forcing. There are also potential positive feedbacks in the carbon cycle, including methane hydrates in the ocean, and peat frozen in permafrost, that are most sensitive to the long tail of the fossil fuel CO2 in the atmosphere.