Nipah virus is a highly virulent zoonotic paramyxovirus causing severe respiratory and neurological disease. Despite its lethality, there is no approved treatment for Nipah virus infection. The viral polymerase complex, composed of the polymerase (L) and phosphoprotein (P), replicates and transcribes the viral RNA genome. Here, we describe structures of the Nipah virus L-P polymerase complex and the L-protein's Connecting Domain (CD). The cryo-electron microscopy L-P complex structure reveals the organization of the RNA-dependent RNA polymerase (RdRp) and polyribonucleotidyl transferase (PRNTase) domains of the L-protein, and shows how the P-protein, which forms a tetramer, interacts with the RdRp-domain of the L-protein. The crystal structure of the CD-domain alone reveals binding of three Mg ions. Modelling of this domain onto an AlphaFold 3 model of an RNA-L-P complex suggests a catalytic role for one Mg ion in mRNA capping. These findings offer insights into the structural details of the L-P polymerase complex and the molecular interactions between L-protein and P-protein, shedding light on the mechanisms of the replication machinery. This work will underpin efforts to develop antiviral drugs that target the polymerase complex of Nipah virus.