The currently ongoing climate change and the debate about possible measures to be taken to limit the consequences of climate change, requires to know and understand the future response of the climate system to greenhouse gas emissions. The Equilibrium Climate Sensitivity (ECS) - defined as the equilibrium global surface temperature response to a doubling of CO$_2$ - is a key predictor of climate change and has been estimated from climate models, observational, historical and palaeoclimate data. However, its distribution (1.5 to $4.5~^\circ$C) as reported in the last IPCC report remains relatively uncertain and has not changed much from the very first estimates. Even more worrying, the latest versions of the most complex Earth System models (CMIP6) suggest even higher estimates for ECS than previous model generations.

This does not mean that climate system science has not advanced; the climate system shows internal variability on many timescales, is subject to non-stationary forcing and is most likely out of equilibrium with the changes in the radiative forcing. Slow and fast feedbacks complicate the interpretation of geological records as feedback strengths vary over time. In the geological past, the forcing timescales were different than at present, suggesting that the response may have behaved differently. Abrupt transitions associated with tipping elements in climate subsystems have occurred in the past and are likely to occur in the future.

In this lecture I will review the progress made in the theoretical understanding of the climate sensitivity. I will introduce the climate attractor and discuss more general notions of ECS on the attractor that can be useful in understanding the response of a climate state to changes in radiative forcing. For example, different time scales in both forcing and response need to be taken into account, and the general underlying assumption of a time-scale separation should be carefully evaluated. The current ECS defines a linear response; however, a climate state close to a tipping point will have a degenerate linear response to perturbations, which can be associated with extreme values of the ECS.