Climate change has evolved from a marginal environmental concern to a systemic risk with significant consequences for global economic and financial stability, requiring a fundamental reassessment of risk management and regulatory frameworks in the financial sector. Physical risks, evident in the rising frequency and severity of extreme weather events, and transition risks, stemming from policy changes such as carbon pricing, technological innovations, and shifting market preferences towards sustainability, present intricate challenges. These risks endanger individual financial institutions and lead to systemic instability, highlighting the necessity for proactive, coordinated responses from financial institutions, regulators, and policymakers. Our research examines the incorporation of climate-related risks into financial stability frameworks through the prism of qualitative and quantitative components. The qualitative component examines global practices via conceptual analysis and framework evaluations (e.g., NGFS scenarios, TCFD reporting, EU SFDR). The quantitative component, assessed through regression and mediation analyses, empirically evaluates the relationships between climate risk indicators and financial stability metrics to comprehensively understand financial vulnerabilities caused by climate change and the effectiveness of mitigation and adaptation strategies. Building on initial regression analysis, we employ mediation analysis to assess the indirect transmission channels through which climate variables influence macroeconomic outcomes. This approach reveals that certain climate indicators, such as renewable energy use and emissions embedded in exports, affect financial stability directly and indirectly through inflationary and fiscal pathways. Our findings offer evidence-based recommendations for improving risk management, developing effective macroprudential instruments, and harmonising climate and financial policy objectives, thus making a substantial contribution to the sustainable finance literature.