Influenza polymerase (FluPol) plays a key role in the viral infection cycle by transcribing and replicating the viral genome. FluPol is a multifunctional, heterotrimeric enzyme with cap-binding, endonuclease, RNA-dependent RNA polymerase and polyadenylation activities. It performs its functions in the context of the viral ribonucleoprotein particle (RNP), wherein the template viral RNA is coated by multiple copies of viral nucleoprotein. Moreover, it interacts with a number of host proteins that are essential cofactors and, consequently, adaptive mutations in the polymerase are required for crossing the avian-human species barrier. In this review, we show how mechanistic understanding of how FluPol performs its multiple functions has greatly advanced over the last decade through determination of high-resolution structures by X-ray crystallography and cryo-electron microscopy. These have revealed not only the detailed architecture of FluPol but highlighted the remarkably conformational flexibility that is inherent to its functioning as a dynamic RNA synthesis machine. Structural studies are also underpinning current attempts to develop next-generation anti-influenza drugs that directly target FluPol.