Mosquito-borne viruses are leading causes of morbidity and mortality in many parts of the world. In recent years, modelling studies have shown that climate change strongly influences vector-borne disease transmission, particularly rising temperatures. As a result, the risk of epidemics has increased, posing a significant public health risk. This review aims to summarize all published laboratory experimental studies carried out over the years to determine the impact of temperature on the transmission of arboviruses by the mosquito vector. Given their high public health importance, we focus on dengue, chikungunya, and Zika viruses, which are transmitted by the mosquitoes Aedes aegypti and Aedes albopictus. Following PRISMA guidelines, 34 papers were included in this systematic review. Most studies found that increasing temperatures result in higher rates of infection, dissemination, and transmission of these viruses in mosquitoes, although several studies had differing findings. Overall, the studies reviewed here suggest that rising temperatures due to climate change would alter the vector competence of mosquitoes to increase epidemic risk, but that some critical research gaps remain.