This article reviews recent scientific progress, relating to four major systems that could exhibit threshold behaviour: ice sheets, the Atlantic meridional overturning circulation (AMOC), tropical forests and ecosystem responses to ocean acidification. The focus is on advances since the Intergovernmental Panel on Climate Change Fifth Assessment Report (IPCC AR5). The most significant developments in each component are identified by synthesizing input from multiple experts from each field. For ice sheets, some degree of irreversible loss (timescales of millennia) of part of the West Antarctic Ice Sheet (WAIS) may have already begun, but the rate and eventual magnitude of this irreversible loss is uncertain. The observed AMOC overturning has decreased from 2004–2014, but it is unclear at this stage whether this is forced or is internal variability. New evidence from experimental and natural droughts has given greater confidence that tropical forests are adversely affected by drought. The ecological and socio-economic impacts of ocean acidification are expected to greatly increase over the range from today’s annual value of around 400, up to 650 ppm CO2 in the atmosphere (reached around 2070 under RCP8.5), with the rapid development of aragonite undersaturation at high latitudes affecting calcifying organisms. Tropical coral reefs are vulnerable to the interaction of ocean acidification and temperature rise, and the rapidity of those changes, with severe losses and risks to survival at 2 °C warming above pre-industrial levels. Across the four systems studied, however, quantitative evidence for a difference in risk between 1.5 and 2 °C warming above pre-industrial levels is limited.