The term “tipping point” has been coined to describe the moment when an observed system suddenly changes state, despite only gradually changing forces or environment. Tipping points are difficult to predict and difficult to reverse. Examples range from capsizing boats to fishery collapse, financial market crash, melting polar ice caps, shifts in ecosystems, and mood changes. Mathematical frameworks for understanding how tipping points can arise as bifurcations have long been in place. More recent research extends these ideas to explore the interaction between tipping and stochasticity in noisy systems. It also aims to identify additional robust mechanisms, outside of the classic bifurcation theory, for threshold behavior that arises in applications. Pressing sustainability questions are now placing the study of tipping points in the context of policy decision support. These are driving efforts to extract, from measurements, indicators of resilience to tipping and early warning signs for proximity to a tipping point. The symposium speakers will discuss these ideas in the context of applications to climate and ecology.