This chapter presents a review of the extant scholarly research focusing on climate risk in banking and its effects on financial stability. The chapter provides suitable guidelines to facilitate a comprehensive understanding of the significance of climate risks for sustainable finance and economic stability. It analyzes the risks that banking institutions face, how financial stability is affected, and the regulations and policies implemented. The extant literature on the subject is also reviewed, with the existing measures used to assess climate risks analyzed. Despite the recent and growing interest in research related to climate risk and its implications for the financial field, the literature is still in its formative stage. We therefore provide an agenda for future research, identifying and discussing the most influential and trending themes that will provide guidance to potential researchers and set future trends in the literature.