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Specific components of the Earth system may abruptly change their state in response to gradual changes in forcing. This possibility has attracted great scientific interest in recent years, and has been recognized as one of the greatest threats associated with anthropogenic climate change. Examples of such components, called tipping elements, include the Atlantic Meridional Overturning Circulation, the polar ice sheets, the Amazon rainforest, as well as the tropical monsoon systems. The mathematical language to describe abrupt climatic transitions is mainly based on the theory of nonlinear dynamical systems and, in particular, on their bifurcations. Applications of this theory to nonautonomous and stochastically forced systems are a very active field of climate research. The empirical evidence that abrupt transitions have indeed occurred in the past stems exclusively from paleoclimate proxy records. In this review, we explain the basic theory needed to describe critical transitions, summarize the proxy evidence for past abrupt climate transitions in different parts of the Earth system, and examine some candidates for future abrupt transitions in response to ongoing anthropogenic forcing. Predicting such transitions remains difficult and is subject to large uncertainties. Substantial improvements in our understanding of the nonlinear mechanisms underlying abrupt transitions of Earth system components are needed. We argue that such an improved understanding requires combining insights from (a) paleoclimatic records; (b) simulations using a hierarchy of models, from conceptual to comprehensive ones; and (c) time series analysis of recent observation-based data that encode the dynamics of the present-day Earth system components that are potentially prone to tipping.