Paleoclimatic data are used in eco-evolutionary models to improve knowledge of biogeographical processes that drive patterns of biodiversity through time, opening windows into past climate-biodiversity dynamics. Applying these models to harmonised simulations of past and future climatic change can strengthen forecasts of biodiversity change. StableClim provides continuous estimates of climate stability from 21,000 years ago to 2100 C.E. for ocean and terrestrial realms at spatial scales that include biogeographic regions and climate zones. Climate stability is quantified using annual trends and variabilities in air temperature and precipitation, and associated signal-to-noise ratios. Thresholds of natural variability in trends in regional- and global-mean temperature allow periods in Earth's history when climatic conditions were warming and cooling rapidly (or slowly) to be identified and climate stability to be estimated locally (grid-cell) during these periods of accelerated change. Model simulations are validated against independent paleoclimate and observational data. Projections of climatic stability, accessed through StableClim, will improve understanding of the roles of climate in shaping past, present-day and future patterns of biodiversity.