More than 13% of the area of the Caucasus glaciers is covered by debris affecting glacier mass balance. Using the Caucasus as example, we introduce a new model configuration that incorporates a physically-based subroutine for the evolution of supraglacial debris into the Global Glacier Evolution Model (GloGEMflow), enabling its application at a regional level. Temporal evolution of debris cover is coupled to glacier dynamics allowing the thickest debris to accumulate in the areas with low velocity. The future evolution of glaciers in the Northern Caucasus is assessed for five Shared Socioeconomic Pathways (SSP) and significance of explicitly incorporating debris-cover formulation in regional glacier modeling is evaluated. Under the more aggressive scenarios, glaciers are projected to disappear almost entirely except on Mount Elbrus, which reaches 5,642 m above sea level, by 2,100. Under the SSP1-1.9 scenario, glacier ice volume stabilizes by 2040. This finding stresses the importance of meeting the Paris Climate Agreement goals and limiting climatic warming to 1.5 °C. We compare evolution of glaciers in the Kuban (more humid western Caucasus) and Terek (drier central and eastern Caucasus) basins. In the Kuban basin, ice loss is projected to proceed at nearly double the rate of that in the Terek basin during the first half of the 21st century. While explicit inclusion of debris cover in modeling leads to a less pronounced projected ice loss, the maximum differences in glacier length, area, and volume occur before 2,100, especially for large valley glaciers diminishing towards the end of the century. These projections show that on average, fraction of debris-covered ice will increase while debris cover will become thinner towards the end of the 21st particularly under the more aggressive scenarios. Overall, the explicit consideration of debris cover has a minor effect on the projected regional glacier mass loss but it improves the representation of changes in glacier geometry locally.