Due to failures in adaptation to global climate change impacts and related risks of natural disasters the safety of human wellbeing, economic sectors, and infrastructure is challenged worldwide both in developing and developed countries. We analyze major features, trends and transformations in societal responses to adverse climate consequences and in global adaptation to escalating disaster risks attributed to climate variability. The focus is on the recently dynamic formation of international system of adaptive governance - an integral segment of global climate policy along with climate mitigation: its formation is based on cross-level coordination, interplay and interaction between its actors, and on combination of context-specific factors from different geographies. Recent shifts in international discourse on climate loss and damage which is among the key indicators in assessment of climate risks and responses are outlined; problems in its incorporation into the climate international regime under the UNFCCC and controversies in global North-South dialogue on climate responsibility, liability and compensation are explored. It is followed by analysis of the recent breakthrough in climate multilateral negotiations underway during the last three decades over the institutional architecture of climate adaptation regime and its mechanisms of international aid for climate risks reduction in the global South. Prospects for performance of the novel international financial mechanism under the UNFCCC, i.e. the multilateral Fund for climate loss and damage in most vulnerable developing countries are discussed, along with its expected contributions to sustainable development.