Iata chief Giovanni Bisignani said the economic impact of the six-day shutdown had been "devastating" and governments "must take their responsibility" and consider ways to compensate the airlines for lost revenue.
Several European airlines, including British Airways, have criticised the extent of the flight bans.
Meanwhile, all of Europe's main air hubs were operating on Wednesday.
Air traffic agency Eurocontrol said it expected some 22,500 out of a normal weekday total of 28,000 flights to go ahead.
"It is anticipated that almost 100% of air traffic will take place in Europe" on Thursday, the agency added.
Britain reopened its airspace from 2200 local time (2100 GMT) on Tuesday, allowing long-haul flights to land at Heathrow airport, Europe's busiest.
Camping at New York's JFK airport
A flight from Vancouver, Canada, was the first to arrive.
Weary passengers cheered and clapped as flights began to take off from airports.
Eve Dickinson, who was among the first to arrive back in Britain, said: "We're absolutely delighted to be home."
British Airways said it would operate all long-haul flights departing from Heathrow and Gatwick airports on Wednesday.
Air traffic controllers in Germany said all restrictions on the country's airspace had been lifted.
Air France said its long-haul flights are now departing as normal, although services in parts of northern Europe remained suspended.
Around the world, airlines began putting on extra flights to clear the backlog of stranded travellers.
Long queues of passengers formed at some of Europe's major airports, including in Paris, Frankfurt and Madrid.
Coaches chartered by the British government have begun ferrying passengers arriving off long-haul flights at Madrid's airport to French ports for the crossing back to the UK.
In Calais there was a stand-off involving two coachloads of Britons who had travelled from Munich after arriving from various European destinations.
The passengers staged a sit-in saying they had been promised full passage to the UK. The stand-off was resolved after the British Foreign Office agreed to pay for the ferry crossing and their transfer by coach to their final destinations.
European Transport Commissioner Siim Kallas has denied the EU had taken too long to respond to calls for airspace to be reopened, saying people's lives were at stake.
The UN's International Civil Aviation Organization (ICAO) said it would lead moves to develop a global standard for the concentration of ash in the air beyond which it was dangerous to fly.
But final decisions about safety would remain up to governments, ICAO Council president Roberto Kobeh Gonzalez said.
The flight ban was imposed because in the high temperatures of an engine turbine, ash can turn to molten glass and cripple the engine.
As travellers scrambled for other modes of transport, ferry and railway companies enjoyed an unexpected bonanza, while some car-hire firms were reportedly increasing charges.
But many other businesses have been hit hard by the chaos.
The carmaker BMW said it was suspending production at three of its plants in Germany because of interruptions in the supply of parts. In Japan, Nissan also suspended production lines, while Honda announced a partial halt to production.
Blocked shipments of goods are also reportedly stacking up in Asian countries including China, South Korea and Bangladesh, while some African exporters of fresh flowers and vegetables are having to throw away tonnes of rotting stock.
Iceland's civil protection agency said the Eyjafjallajokull volcano had lost nearly 80% of its intensity since the weekend, although the situation remains changeable.
There are concerns the eruption could set off the nearby, larger Katla volcano, which sits on the Myrdalsjokull glacier, but officials said no activity had been detected. Its last major eruption was in 1918.
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