Flaviviruses are vector-borne RNA viruses, many of which are clinically relevant human viral pathogens, such as dengue, Zika, Japanese encephalitis, West Nile and yellow fever viruses. Millions of people are infected with these viruses around the world each year. Vaccines are only available for some members of this large virus family, and there are no effective antiviral drugs to treat flavivirus infections. The unmet need for vaccines and therapies against these flaviviral infections drives research towards a better understanding of the epidemiology, biology and immunology of flaviviruses. In this review, we discuss the basic biology of the flavivirus replication process and focus on the molecular aspects of viral genome replication. Within the virus-induced intracellular membranous compartments, flaviviral RNA genome replication takes place, starting from viral poly protein expression and processing to the assembly of the virus RNA replication complex, followed by the delivery of the progeny viral RNA to the viral particle assembly sites. We attempt to update the latest understanding of the key molecular events during this process and highlight knowledge gaps for future studies.