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The courts have become an increasingly important venue for efforts to combat the impacts of climate change. Claims have brought on a range of legal grounds, but many climate lawsuits are based on the existence of a causal relationship between emissions of greenhouse gases and impacts that occur as a result. Such claims may be made to establish emitters' liability for climate change impacts, or to demonstrate that contributions to climate change cause harm, and therefore that greenhouse gas emissions should be curbed. Establishing causation in climate litigation requires synthesis of climate science and legal arguments. This thesis evaluates how legal arguments can align with developments in climate change attribution science, a subfield of climate science that can assess the contribution of anthropogenic influence to climate-related impacts. It then considers how attribution-science evidence has been used in past lawsuits, and responds to emerging scientific questions regarding the effects of climate change that have arisen in the courts. Legal scholars have identified attribution science as holding the potential to establish causation in the courts, but the analyses described in section 4 of this thesis found that the evidence presented in climate lawsuits often lags developments in climate science and that, in some cases, this may have hindered findings of specific causal relationships between greenhouse gases and their consequences. Responding to these findings, sections 5 and 6 report analyses that advance the state of knowledge in areas of attribution science that hold particular legal relevance. Section 5 studied anthropogenic influence on the glacial lake outburst flood hazard from Laguna Palcacocha in the Peruvian Andes, in the context of ongoing litigation (Lliuya v. RWE). It concluded that 85-105% (5-95% confidence interval) of the observed industrial-era warming in the region surrounding Laguna Palcacocha was attributable to anthropogenic climate change and that the overall retreat of the Palcaraju glacier since 1880 is entirely attributable to this warming. Moreover, it is virtually certain (>99% probability) that the observed glacier retreat cannot be explained by natural variability alone. Laguna Palcacocha has expanded in the wake of this glacier retreat, substantially increasing the outburst flood hazard to the downstream city of Huaraz. Finally, this thesis responds to the increased use of human rights-based arguments in climate litigation and develops methods that extend attribution science analyses from quantifying the effect of climate change on physical hazards to their impacts on health. These methods address limitations in existing approaches, including the representation of the climate-health relationship, choices in calculating counterfactual temperatures, differing approaches for assessing long-term trends and individual events, and estimating the effects of adaptation. The approach described in this section is then applied to quantify heat-related mortality attributable to climate change in the Canton of Zürich (Switzerland) over 1969-2018. We found a substantial burden of heat-related mortality attributable to climate change both within and outside of heatwaves. Overall, 1,7000 heat-related deaths were attributed to climate change, and adaptation was found to have avoided at least a further 700 deaths. The thesis spans climate science, glaciology, epidemiology, and legal analysis. Altogether, this research explains how science can strengthen legal causation arguments in climate litigation, and develops methods and analyses to this end. Scientific evidence has a central role to play in developing well-grounded climate lawsuits, and methodological developments can resolve questions raised in court.