Airlines said that Heathrow, Europe's busiest airport, was the worst hit by the poor weather - with only a limited number of flights to arrive and depart. Hundreds of passengers camped overnight in terminal buildings after services were canceled or delayed.
British Airways said all short-haul flights from Heathrow would be canceled Monday.
Eurostar reported that its trains linking England to France and Belgium were also severely delayed or canceled and urged travelers to cancel or postpone their trips if possible. Lines of delayed passengers snaked outside the St. Pancras rail station in central London.
Criticism of the situation at London's Heathrow and other airports prompted British officials to promise an investigation.
In Germany flight operations were slowed even though Frankfurt airport, Germany's biggest, was clear of snow and ice. Officials canceled about 300 flights there Monday, out of a planned total of 1,340, because of problems elsewhere in Europe, airport operator Fraport said.
French civil aviation authorities, meanwhile, asked airlines to reduce their flights at the two main Paris airports by 30 percent.
French aviation spokesman Eric Heraud said these reductions take into account the problems at other European airports that may be destinations for flights from Charles de Gaulle or Orly airports in Paris.
More snow is forecast in some areas of Britain for Monday afternoon, adding to the problems, with British Airways warning of more flight cancellations, particularly in the greater London area, where all airports have been affected.
Winter storms forced British government ministers and bank executives to postpone their meeting Monday on the politically touchy issue of bank bonuses. The Department of Business, Innovation and Skills did not announce a new date but said it hoped the meeting could be rescheduled later this week.
Airports and British travel industry group ABTA have warned it is almost inevitable that some cancellations and delays will continue through this week and likely snarl those attempting to head away for the holiday season.
British Airways warned passengers not to travel to London's Heathrow airport unless they have a confirmed seat on a flight known to be operating despite the weather problems and the backlog of delays. It urged travelers to consider canceling their flight if possible.
Icy conditions were also hampering travel across Europe, with flights canceled and delayed in multiple countries at the weekend.
"We are awaiting further updates," before publishing the schedule for the remainder of the day, British Airways said in a statement. "Customers should continue to check their flight status as more snow is forecast this afternoon, which could cause further disruption to airport operations."
In Britain criticism of the breakdown in the air and road system mounted, prompting Transport Secretary Philip Hammond to promise an inquiry into the way passengers were treated after their flights were canceled. He was expected to address Parliament about the crisis Monday afternoon.
Travelers described scenes of chaos for those arriving at the Heathrow airport, with officials offering contradictory messages about the status of flights.
At Heathrow's sprawling Terminal 5, tired and disgruntled passengers faced lengthy waits without much information.
American Suzie Devoe, 20, was one of many who had spent two nights sleeping on the airport floor in a bid to get home for the holidays. She was desperately trying to rearrange a flight so she could get back to Washington to spend Christmas with her family. "The whole situation has been a complete nightmare," said the Bristol University student. "I just want to get home, I want to be with my family. But I'm being held in a horrible limbo."
In France, Jean Louis Balam, a Dutch passenger who spent the night at Charles de Gaulle airport trying for a second day to get from Paris to Amsterdam said passengers had to improvise overnight at the airport.
"We went to the airport yesterday evening and we wanted to go to Amsterdam and we waited here about five hours," he told Associated Press Television News. "We had to sleep at the airport because ... hotels were full. "
Blandine Sabadie also found herself sleeping at the airport. She said passengers were escorted to an "improvised" area with portable beds, blankets and warm drinks.
French Transport Minister Thierry Mariani said on France-Info radio that it is "unacceptable" that some 3,000 people were blocked at Charles de Gaulle airport over the weekend and called a meeting this week of airlines to find ways to improve communication with passengers.
The minister, explaining multiple delays around Europe, said that when a runway is closed for an hour the lost time cannot be reclaimed. "For each hour lost, it is some 70 to 80 flights that you can't recover during the day," he said.
Authorities in parts of western Germany banned trucks weighing more than 7.5 tons from highways as a safety measure.
In the Czech Republic eight regional railways were shut because of the weather but the international airport at Prague remained open.
Bus traffic in the Paris region was "very disturbed," the RATP transport authority said, with buses to the airports halted.
Forecasters have said Britain is experiencing some of the most severe winter weather in a century, with continued freezing temperatures and snowfall accumulations expected Monday afternoon and evening.
Temperatures plunged to a record low overnight in Northern Ireland, and forecasters predicted fresh deluges of snow across Britain.
Motoring organizations have warned of potentially fatal conditions on Britain's icy roads.
Geir Moulson in Berlin and Elaine Ganley in Paris contributed to this report.