Transmission of arthropod-borne viruses (arboviruses) involves infection and replication in both arthropod vectors and vertebrate hosts. Nearly all arboviruses are RNA viruses with high mutation frequencies, which leaves them vulnerable to genetic drift and fitness losses owing to population bottlenecks during vector infection, dissemination from the midgut to the salivary glands and transmission to the vertebrate host. However, despite these bottlenecks, they seem to avoid fitness declines that can result from Muller's ratchet. In addition, founder effects that occur during the geographic introductions of human-amplified arboviruses, including chikungunya virus and Zika virus, can affect epidemic and endemic circulation, as well as virulence. In this Review, we discuss the role of genetic drift following population bottlenecks and founder effects in arboviral evolution and spread, and the emergence of human disease.