Some emerging technologies promise to significantly improve the human condition, but come with a risk of failure so catastrophic that human civilization may not survive. This article discusses the great downside dilemma posed by the decision of whether or not to use these technologies. The dilemma is: use the technology, and risk the downside of catastrophic failure, or do not use the technology, and suffer through life without it. Historical precedents include the first nuclear weapon test and messaging to extraterrestrial intelligence. Contemporary examples include stratospheric geoengineering, a technology under development in response to global warming, and artificial general intelligence, a technology that could even take over the world. How the dilemma should be resolved depends on the details of each technology's downside risk and on what the human condition would otherwise be. Meanwhile, other technologies do not pose this dilemma, including sustainable design technologies, nuclear fusion power, and space colonization. Decisions on all of these technologies should be made with the long-term interests of human civilization in mind. This paper is part of a series of papers based on presentations at the Emerging Technologies and the Future of Humanity event held at the Royal Swedish Academy of Sciences on 17 March 2014.