Through annual epidemics and global pandemics, influenza A viruses (IAVs) remain a significant threat to human health as the leading cause of severe respiratory disease. Within the last century, four global pandemics have resulted from the introduction of novel IAVs into humans, with components of each originating from avian viruses. IAVs infect many avian species wherein they maintain a diverse natural reservoir, posing a risk to humans through the occasional emergence of novel strains with enhanced zoonotic potential. One natural barrier for transmission of avian IAVs into humans is the specificity of the receptor-binding protein, hemagglutinin (HA), that recognizes sialic acid-containing glycans on host cells. HAs from human IAVs exhibit "human-type" receptor specificity, binding exclusively to glycans on cells lining the human airway where terminal sialic acids are attached in the α2-6 configuration (NeuAcα2-6Gal). In contrast, HAs from avian viruses exhibit specificity for "avian-type" α2-3-linked (NeuAcα2-3Gal) receptors, and thus require adaptive mutations to bind human-type receptors. Since all human IAV pandemics can be traced to avian origins, there remains ever-present concern over emerging IAVs with human-adaptive potential that might lead to the next pandemic. This concern has been brought into focus through emergence of SARS-CoV-2, aligning both scientific and public attention to the threat of novel respiratory viruses from animal sources. In this review we summarize receptor-binding adaptations underlying the emergence of all prior IAV pandemics in humans, maintenance and evolution of human-type receptor specificity in subsequent seasonal IAVs, and potential for future human-type receptor adaptation in novel avian HAs.