Previous influenza A virus (IAV) pandemics have invariably been caused by the introduction of an emergent IAV strain from an animal host into a human population with no or only little pre-existing immunity to the novel strain. Although zoonotic spillover of IAVs into humans can be associated with severe disease and a high fatality rate, these strains are typically poorly adapted to humans and are unable to establish sustained transmission between humans. Given the presumably very high degree of exposure to animal populations with endemic IAV, the number of pandemics remains surprisingly low. In this review, we provide an updated perspective on the molecular mechanisms underlying the adaptation of zoonotic IAV to human hosts, and discuss the implications for future pandemics.