This study examines the integration of climate risks into both Pillar 1 and Pillar 2 frameworks, focusing on the main approaches and tools that institutions and supervisors are using to address these risks more effectively. We shed light on these approaches by adding the perspective of risk experts, highlighting which methodologies are currently considered practical and implementable, and identifying the key challenges institutions face in conducting this integration. We also situate these approaches within the broader regulatory landscape, highlighting the unique characteristics of the approaches required to manage climate risks compared to other risk types and drivers.