Air transport is energy-intense, and considerable attention has been paid to the sector's use of fuel and emissions of greenhouse gases. Commercial aviation is believed to currently emit about 1 Gt CO2 per year, if considering global bunker fuel use (scope 1 in the Greenhouse Gas Protocol). A growing database is becoming available on scope 1-3 emissions; this is, including up- and downstream emissions, and it is now possible to assess the aviation system's carbon intensity more comprehensively. This paper investigates the annual reports of 26 of the largest airlines in the world by market capitalisation, finding that reporting on emissions for scopes 1-3 is still inconsistent and characterised by reporting gaps. Yet, available data suggests that scope 3 emissions are significant (about 30% of scope 1 emissions). These findings have repercussions for the sector's net-zero ambitions, climate governance, consumer choices and air transport finance, as the overall contribution from air travel to climate change remains underestimated. Results suggest that it is in the sector's interest to present robust, transparent, consistent and accurate emission inventories - and to engage with the implications.