Airports in London will remain closed until at least Wednesday, though British Airways sought permission to land about a dozen flights from the United States, Asia and Africa at Heathrow Airport. The flights are en route and may have to head to airports elsewhere.
Still, it was the first day since Iceland's Eyjafjallajokull volcano erupted Wednesday that travelers were given a glimmer of hope.
Cheers and applause broke out as flights took off from Paris' Charles de Gaulle Airport, Amsterdam and elsewhere. German airspace also remained officially closed but 800 planes were allowed to land or take off, all flying at low altitude.
"Everyone was screaming in the airplane from happiness," said Savvas Toumarides of Cyprus, who arrived in New York after getting stuck in Amsterdam for five days and missing his sister's wedding. He said the worst part was "waiting and waiting and not knowing."
"We were in the hotel having breakfast, and we heard an aircraft take off. Everybody got up and applauded," said Bob Basso of San Diego, who has been stranded near Charles de Gaulle since Friday.
The Eurocontrol air traffic agency in Brussels said it expected just under half of Europe's 27,500 flights to go ahead Tuesday, a marked improvement over the last few days. The agency predicted close to normal takeoffs by Friday.
"The situation today is much improved," said Brian Flynn, deputy head of operations at the Brussels-based agency.
But with more than 95,000 flights canceled in the last week alone, airlines faced the enormous task of working through the backlog to get passengers where they want to go -- a challenge that could take days or even weeks.
Passengers with current tickets were being given priority -- stranded passengers were being told to either pay for a new ticket, take the first available flight or to use their old ticket and wait for days, or weeks, for the first available seat.
"I'm supposed to be home, my children are supposed to be in school," said Belgian Marie-Laurence Gregoire, 41, who was traveling in Japan with her husband and three children, ages 6, 8, 10. They said the best that British Airways could do was put them on a flight to Rome.
"I'm tired. I just want to go home," she said.
Although seismic activity at the volcano has increased, the ash plume appeared to be shrinking Tuesday. Still, scientists were worried that the activity could trigger an even larger eruption at the nearby Katla volcano, which sits on the massive Myrdalsjokull icecap and has erupted every 80 years or so -- the last time in 1918.
"The activity of one volcano sometimes triggers the next one, and Katla has been active together with Eyjafjallajokull in the past," said Pall Einarsson, professor of geophysics at the Institute of Earth Sciences at the University of Iceland.
At eruption at Katla could spark similar travel disruptions, depending on the prevailing winds. But in Iceland's eight volcanic eruptions in the last 40 years, only the recent one at Eyjafjallajokull was followed by winds blowing toward northern Europe.
An international pilots group warned of continued danger because of the ash, which drifted over the North Sea and was being pushed back over Britain on Tuesday by shifting north winds.
A Eurocontrol volcanic ash map on Tuesday listed the airspace between Iceland, Britain and Ireland as a no-fly zone, along with much of the area around the Baltic Sea. The ash cloud also spread westward from Iceland, toward Greenland and Canada's eastern coastline.
Still, planes were being allowed to fly above 20,000 feet (7,000 kilometers) over the United Kingdom.
Herbert Puempel at the World Meteorological Organization in Geneva said there was a small possibility that some far-flung airports on the Canadian east coast, such as Goose Bay or Thunder Bay, might be affected by the ash but said "a serious effect on the eastern seaboard I think is very unlikely."
The volcano was also grumbling -- tremors, which geologists believe to be caused by magma rising through the crust, can be heard and felt as far as 16 miles (25 kilometers) from the crater.
"It's like a shaking in the belly. People in the area are disturbed by this," said Kristin Vogfjord, geologist at the Icelandic Met Office.
Scottish airports let in a handful of domestic flights, while Switzerland and northern Italy also opened their airspace. Some flights took off from Asia to southern Europe and came in from Cairo, where at least 17,000 people had been stranded.
Airports in central Europe and Scandinavia were open and most of southern Europe remained clear. Spain volunteered to be an emergency hub for overseas travelers trying to get home and piled on extra buses, trains and ferries to handle the expected crush.
Britain sent a navy ship to Spain to fetch 500 troops coming home from Afghanistan and pick up hundreds of passengers stranded by the chaos.
"How many modes of transport have I been on? I have lost count now," said Angus Henderson, 40, of the 1st Battalion, Royal Welsh, an infantry unit. "Planes, buses and now ships."
Henderson was pressing to get back to Britain to see his wife and three small kids and attend the funeral of a colleague killed in Afghanistan. But the trip on the HMS Albion, a 570-foot (173-meter-long) amphibious assault ship, will take 40 hours from Santander in northern Spain to Portsmouth, England.
Patricia Quirke of Manchester said she and nine other families drove all night across Spain just to catch the Royal Navy ride.
Many Asian airports and airlines remained cautious, and most flights to and from Europe were still canceled. Australia's Qantas canceled its Wednesday and Thursday flights from Asia to Frankfurt and London, as well as return flights to Asia, saying the situation was too uncertain.
The aviation industry -- facing losses of more than $1 billion -- has sharply criticized European governments' handling of the disruption that grounded thousands of flights on the continent. But its first order of business was to cut down that flight backlog.
"We've never had a backlog like this before," said Laurie Price, director of aviation strategy at consultant Mott Macdonald. Spain's main airline Iberia said it was using bigger planes and adding extra flights to help stranded passengers get to their destinations. Other airlines were hiring buses to help customers get home.
Most airlines said they would let passengers with tickets for a departing flight this week go first, but offered to rebook customers on another plane for no additional cost.
British Airways, which has canceled about 500 flights a day for the past five days, said it was trying to clear its backlog on a case-by-case basis. It said travelers could either rebook online or claim a full refund, and it also urged travelers booked to fly this week to consider canceling their trips so the airline could fly more people home.
In the end, many people did not get a flight out Tuesday.
Phil Livingstone, a university student from St. Helens, England, spent three nights sleeping on chairs and eating cups of noodles at Seoul's Incheon International Airport.
"Hope is high at the minute just because it's the only thing we've got," he said.