How influenza A viruses host-jump from animal reservoir species to humans, which can initiate global pandemics, is a central question in pathogen evolution. The zoonotic and spatial origins of the influenza virus associated with the “Spanish flu” pandemic of 1918 have been debated for decades. Outbreaks of respiratory disease in US swine occurred concurrently with disease in humans, raising the possibility that the 1918 virus originated in pigs. Swine also were proposed as “mixing vessel” intermediary hosts between birds and humans during the 1957 Asian and 1968 Hong Kong pandemics. Swine have presented an attractive explanation for how avian viruses overcome the substantial evolutionary barriers presented by different cellular environments in humans and birds. However, key assumptions underpinning the swine mixing-vessel model of pandemic emergence have been challenged in light of new evidence. Increased surveillance in swine has revealed that human-to-swine transmission actually occurs far more frequently than the reverse, and there is no empirical evidence that swine played a role in the emergence of human influenza in 1918, 1957, or 1968. Swine-to-human transmission occurs periodically and can trigger pandemics, as in 2009. But swine are not necessary to mediate the establishment of avian viruses in humans, which invites new perspectives on the evolutionary processes underlying pandemic emergence.