Influenza A viruses (IAVs) typically cause a mild to moderate respiratory disease, whereas infections with pandemic and highly pathogenic avian IAV strains are frequently associated with high morbidity and death. Various noncanonical or aberrant transcription and replication products have been implicated in the effect of IAV infection on disease outcomes. While early research indicated that all these molecules may be defective, recent findings coupled with analyses of the structure of the IAV RNA polymerase suggest that the production of noncanonical RNAs is not solely driven by errors. Instead, their place in infection may be more nuanced. In this review, we discuss our current understanding of the molecular steps that underlie noncanonical transcription and replication and which molecular mysteries remain.