The influenza virus polymerase transcribes or replicates the segmented RNA genome (viral RNA) into viral messenger RNA or full-length copies. To initiate RNA synthesis, the polymerase binds to the conserved 3' and 5' extremities of the viral RNA. Here we present the crystal structure of the heterotrimeric bat influenza A polymerase, comprising subunits PA, PB1 and PB2, bound to its viral RNA promoter. PB1 contains a canonical RNA polymerase fold that is stabilized by large interfaces with PA and PB2. The PA endonuclease and the PB2 cap-binding domain, involved in transcription by cap-snatching, form protrusions facing each other across a solvent channel. The 5' extremity of the promoter folds into a compact hook that is bound in a pocket formed by PB1 and PA close to the polymerase active site. This structure lays the basis for an atomic-level mechanistic understanding of the many functions of influenza polymerase, and opens new opportunities for anti-influenza drug design.