BRUSSELS - As New Year revelers across the world welcomed 2013 with lavish fireworks and parties, leaders in debt-plagued Europe were in a more restrained mood - hopeful about the recovery of their battered economies but cautious about the bumpy road ahead.

"The reforms we have agreed to are beginning to take effect," German Chancellor Angela Merkel said in her New Year's address.

But she warned that the eurozone debt crisis is "far from over" and the economic environment of next year will be more difficult.

"That should not discourage us. It should spur us on," Merkel said, noting that the three-year-old sovereign debt crisis shows how important it is to strike a balance between prosperity and solidarity.

"We still need a lot of patience," she added.

In eurozone's second largest economy, French President Francois Hollande, whose popularity suffers a severe setback due to economic woes and social strains, admitted that French people's "concerns are legitimate" and "difficulties are serious."

But in his first New Year's televised address at the Elysee Palace, he said France would emerge from the crisis "sooner and stronger" than expected because of the policies of his government.

"We've set the course - jobs, competitiveness and growth - and I will not deviate. It's the future of France," he said.

British Prime Minister David Cameron marked the New Year by hailing an "extraordinary" 2012 and calling for "realism and optimism" for the future.

"We are still dealing with debts that built up over many years. And for many families, making ends meet is difficult," Cameron said, referring to the economic struggle Britain has been in with other European countries.

But he assured the people by saying, "We are on the right track. On all the big issues that matter to Britain, we are heading in the right direction."

People of his country should be realistic because there are no "quick fixes" and "we can be optimistic too because we are making tangible progress," he said.

Even in Greece, where the debt crisis was born and whose economy is projected to suffer a sixth straight year of recession, leaders were not downbeat.

"I am absolutely certain that we can reverse the current situation... that this adventure will end," Greek President Karolos Papoulias said in his message for New Year's Eve.

While acknowledging that the new year would be tough, Greek Prime Minister Antonis Samaras voiced optimism that, with unity, Greeks will leave well back in the past the threat of chaotic default and possible exit from the euro, and move ahead on the path of recovery and prosperity again.

NEW YEAR'S RESOLUTION

Although hopeful about the future, the leaders were sober about arduous tasks ahead.

Merkel said more needs to be done internationally to monitor the financial markets, warning that the world has not sufficiently learned the lesson of the devastating financial crisis of 2008.

Domestically, she emphasized research as a priority and pledged to invest more in education, build Germany into one of the most modern energy producers in the world, prepare the country for demographic change and bring public finances into order.

For Hollande, the "sole objective" in 2013 is to reverse the country's rising unemployment levels.

"All our forces will be mobilized for a sole objective: to reverse the unemployment curve within a year. We will reach that at all costs," he said.

He said the French wealthy have to help more in restoring public finances.

After the country's Constitutional Council rejected a proposed 75-percent upper tax on annual incomes above 1 million euros (1.3 million U.S. dollars), the Socialist president said the law would be redesigned for the same objective.

In Cameron's message to his country, he argued that changes to welfare were necessary in order to get more people into employment and help those who "want to work hard and get on in life," defending his decision to ban the jobless who refuse to try to find a job from claiming the dole.

Italian President Giorgio Napolitano said his country could recover from recession and boost its economy only with "a stronger push and credible proposals for further integration" in the EU.

Spanish King Juan Carlos also said in his New Year message that Spain would continue to work with the EU "to overcome purely national visions" and strengthen the foundations of solidarity.