The potential for nonlinear ‘surprises’ in the climate system has been recognised since the 1960s. One such nonlinearity is a tipping point: when change in a system becomes self-sustaining once forced passed a threshold, triggering a state shift that is often abrupt and irreversible. Research on climate tipping points has flourished since 2005, recently culminating in the release of the Global Tipping Points Report at COP28 and calls for targeted assessment by the Intergovernmental Panel on Climate Change. Here I summarise progress in our growing understanding of climate tipping points and reflect on their implications for climate action and the outlook for future research.