Influenza A viruses cause pandemics when they cross between species and an antigenically novel virus acquires the ability to infect and transmit between these new hosts. The timing of pandemics is currently unpredictable but depends on ecological and virological factors. The host range of an influenza A virus is determined by species-specific interactions between virus and host cell factors. These include the ability to bind and enter cells, to replicate the viral RNA genome within the host cell nucleus, to evade host restriction factors and innate immune responses and to transmit between individuals. In this Review, we examine the host barriers that influenza A viruses of animals, especially birds, must overcome to initiate a pandemic in humans and describe how, on crossing the species barrier, the virus mutates to establish new interactions with the human host. This knowledge is used to inform risk assessments for future pandemics and to identify virus-host interactions that could be targeted by novel intervention strategies.