Flaviviruses are emerging pathogens of increasingly important public health concern in the world. For most flaviviruses such as dengue virus (DENV) and West Nile virus (WNV) neither vaccine nor antiviral treatment is available. The viral RNA-dependent RNA polymerase (RdRp) non-structural protein 5 (NS5) has no equivalent in the host cell and is essential for viral replication. Here, we give an overview of the current knowledge regarding Flavivirus RdRp function and structure as it represents an attractive target for drug design. Flavivirus RdRp exhibits primer-independent activity, thus initiating RNA synthesis de novo. Following initiation, a conformational change must occur to allow the elongation process. Structure-function studies of Flavivirus RdRp are now facilitated by the crystal structures of DENV (serotype 3) and WNV RdRp domains. Both adopt a classic viral RdRp fold and present a closed pre-initiation conformation. The so-called priming loop is thought to provide the initiation platform stabilizing the de novo initiation complex. A zinc-ion binding site at the hinge between two subdomains might be involved in opening up the RdRp structure towards a conformation for elongation. Using two different programs we predicted common potential allosteric inhibitor binding sites on both structures. We also review ongoing approaches of in vitro and cell-based screening programs aiming at the discovery of nucleosidic and non-nucleosidic inhibitors targeting Flavivirus RdRps.