The long-term global mean temperature depends on the total amount of anthropogenic CO2 emitted. This direct link between temperature and cumulative CO2 emissions has implications for policymakers, as the cumulative emissions framework identifies the total amount of carbon that can be emitted, referred to as a cumulative carbon budget, that is consistent with reaching stabilization of the global mean temperature at desired levels, such as 1.5 °C or 2.0 °C warming above the pre-industrial level. This dissertation is a compilation of three studies that explore the relationship between warming and cumulative carbon emissions at high amounts of total carbon emitted (Project I; Chapter 2), its sensitivity to non-CO2 forcing (Project II; Chapter 3), and constraining the climate model responses with observations, in order to provide more accurate estimates of the carbon budget consistent with 1.5 °C warming above the pre-industrial level (Project III; Chapter 4). A joint summary of the key findings from each project, and their significance, is presented in Chapter 5. / Graduate / 2018-05-29