The threat of so-called rapid or abrupt climate change has generated considerable public interest because of its potentially significant impacts. The collapse of the North Atlantic Thermohaline Circulation or the West Antarctic Ice Sheet, for example, would have potentially catastrophic effects on temperatures and sea level, respectively. But how likely are such extreme climatic changes? Is it possible actually to estimate likelihoods? This article reviews the societal demand for the likelihoods of rapid or abrupt climate change, and different methods for estimating likelihoods: past experience, model simulation, or through the elicitation of expert judgments. The article describes a survey to estimate the likelihoods of two characterizations of rapid climate change, and explores the issues associated with such surveys and the value of information produced. The surveys were based on key scientists chosen for their expertise in the climate science of abrupt climate change. Most survey respondents ascribed low likelihoods to rapid climate change, due either to the collapse of the Thermohaline Circulation or increased positive feedbacks. In each case one assessment was an order of magnitude higher than the others. We explore a high rate of refusal to participate in this expert survey: many scientists prefer to rely on output from future climate model simulations.