The ultimate climate emergency is a 'runaway greenhouse': a hot and water-vapour-rich atmosphere limits the emission of thermal radiation to space, causing runaway warming. Warming ceases only after the surface reaches approximately 1400 K and emits radiation in the near-infrared, where water is not a good greenhouse gas. This would evaporate the entire ocean and exterminate all planetary life. Venus experienced a runaway greenhouse in the past, and we expect that the Earth will in around 2 billion years as solar luminosity increases. But could we bring on such a catastrophe prematurely, by our current climate-altering activities? Here, we review what is known about the runaway greenhouse to answer this question, describing the various limits on outgoing radiation and how climate will evolve between these. The good news is that almost all lines of evidence lead us to believe that is unlikely to be possible, even in principle, to trigger full a runaway greenhouse by addition of non-condensible greenhouse gases such as carbon dioxide to the atmosphere. However, our understanding of the dynamics, thermodynamics, radiative transfer and cloud physics of hot and steamy atmospheres is weak. We cannot therefore completely rule out the possibility that human actions might cause a transition, if not to full runaway, then at least to a much warmer climate state than the present one. High climate sensitivity might provide a warning. If we, or more likely our remote descendants, are threatened with a runaway greenhouse, then geoengineering to reflect sunlight might be life's only hope. Injecting reflective aerosols into the stratosphere would be too short-lived, and even sunshades in space might require excessive maintenance. In the distant future, modifying Earth's orbit might provide a sustainable solution. The runaway greenhouse also remains relevant in planetary sciences and astrobiology: as extrasolar planets smaller and nearer to their stars are detected, some will be in a runaway greenhouse state.