Nipah virus (NiV) is a non-segmented negative-strand RNA virus (nsNSV) with high pandemic potential, as it frequently causes zoonotic outbreaks and can be transmitted from human to human. Its RNA-dependent RNA polymerase (RdRp) complex, consisting of the L and P proteins, carries out viral genome replication and transcription and is therefore an attractive drug target. Here, we report cryo-EM structures of the NiV polymerase complex in the apo and in an early elongation state with RNA and incoming substrate bound. The structure of the apo enzyme reveals the architecture of the NiV L-P complex, which shows a high degree of similarity to other nsNSV polymerase complexes. The structure of the RNA-bound NiV L-P complex shows how the enzyme interacts with template and product RNA during early RNA synthesis and how nucleoside triphosphates are bound in the active site. Comparisons show that RNA binding leads to rearrangements of key elements in the RdRp core and to ordering of the flexible C-terminal domains of NiV L required for RNA capping. Taken together, these results reveal the first structural snapshots of an actively elongating nsNSV L-P complex and provide insights into the mechanisms of genome replication and transcription by NiV and related viruses.