Bunyaviruses are negative sense, single-strand RNA viruses that infect a wide range of vertebrate, invertebrate and plant hosts. WHO lists three bunyavirus diseases as priority diseases requiring urgent development of medical countermeasures highlighting their high epidemic potential. While the viral large (L) protein containing the RNA-dependent RNA polymerase is a key enzyme in the viral replication cycle and therefore a suitable drug target, our knowledge on the structure and activities of this multifunctional protein has, until recently, been very limited. However, in the last few years, facilitated by the technical advances in the field of cryogenic electron microscopy, many structures of bunyavirus L proteins have been solved. These structures significantly enhance our mechanistic understanding of bunyavirus genome replication and transcription processes and highlight differences and commonalities between the L proteins of different bunyavirus families. Here, we provide a review of our current understanding of genome replication and transcription in bunyaviruses with a focus on the viral L protein. Further, we compare within bunyaviruses and with the related influenza virus polymerase complex and highlight open questions.