Forecasting the impacts of climate change on Aedes-borne viruses-especially dengue, chikungunya, and Zika-is a key component of public health preparedness. We apply an empirically parameterized model of viral transmission by the vectors Aedes aegypti and Ae. albopictus, as a function of temperature, to predict cumulative monthly global transmission risk in current climates, and compare them with projected risk in 2050 and 2080 based on general circulation models (GCMs). Our results show that if mosquito range shifts track optimal temperature ranges for transmission (21.3-34.0°C for Ae. aegypti; 19.9-29.4°C for Ae. albopictus), we can expect poleward shifts in Aedes-borne virus distributions. However, the differing thermal niches of the two vectors produce different patterns of shifts under climate change. More severe climate change scenarios produce larger population exposures to transmission by Ae. aegypti, but not by Ae. albopictus in the most extreme cases. Climate-driven risk of transmission from both mosquitoes will increase substantially, even in the short term, for most of Europe. In contrast, significant reductions in climate suitability are expected for Ae. albopictus, most noticeably in southeast Asia and west Africa. Within the next century, nearly a billion people are threatened with new exposure to virus transmission by both Aedes spp. in the worst-case scenario. As major net losses in year-round transmission risk are predicted for Ae. albopictus, we project a global shift towards more seasonal risk across regions. Many other complicating factors (like mosquito range limits and viral evolution) exist, but overall our results indicate that while climate change will lead to increased net and new exposures to Aedes-borne viruses, the most extreme increases in Ae. albopictus transmission are predicted to occur at intermediate climate change scenarios.