The ranges of plants and animals are moving in response to recent changes in climate. As temperatures rise, ecosystems with `nowhere to go', such as mountains, are considered to be more threatened. However, species survival may depend as much on keeping pace with moving climates as the climate's ultimate persistence. Here we present a new index of the velocity of temperature change (km yr-1), derived from spatial gradients (°C km-1) and multimodel ensemble forecasts of rates of temperature increase (°Cyr-1) in the twenty-first century. This index represents the instantaneous local velocity along Earth's surface needed to maintain constant temperatures, and has a global mean of 0.42kmyr-1 (A1B emission scenario). Owing to topographic effects, the velocity of temperature change is lowest in mountainous biomes such as tropical and subtropical coniferous forests (0.08kmyr-1), temperate coniferous forest, and montane grasslands. Velocities are highest in flooded grasslands (1.26kmyr-1), mangroves and deserts. High velocities suggest that the climates of only 8% of global protected areas have residence times exceeding 100years. Small protected areas exacerbate the problem in Mediterranean-type and temperate coniferous forest biomes. Large protected areas may mitigate the problem in desert biomes. These results indicate management strategies for minimizing biodiversity loss from climate change. Montane landscapes may effectively shelter many species into the next century. Elsewhere, reduced emissions, a much expanded network of protected areas, or efforts to increase species movement may be necessary.