Presidents and prime ministers were more blunt about the U.N. system at the three-day gathering, which started Wednesday, to mark the United Nations' 60th anniversary.
"If member countries want the United Nations to be respected and effective, they should begin by making sure it is worthy of respect," President Bush said.
"The United Nations should live up to its name," Britain's Prime Minister Tony Blair said.
Instead of a celebration of U.N. achievements since its founding in the ashes of World War II, the summit was much more a somber reappraisal of its shortcomings and a debate about how to meet the daunting challenges of a world where poverty and violence are still endemic.
On Thursday, leaders from countries including Russia, Iraq, China, Israel, France, Venezeula and Afghanistan were to address the General Assembly. British Prime Minister Tony Blair was to host a presentation on poverty reduction, attended by Nigerian President Olesegun Obasanjo and Irish rocker and anti-poverty campaigner Bob Geldof. And a Korean Society dinner was to honor Bush and South Korean President Roh Moo-Hyun.
Coming into the summit, diplomats had to dilute a document on goals for tackling rights abuses, terrorism and U.N. reform because they couldn't settle their disputes.
In opening the summit that he called a year ago in hopes of winning approval for an ambitious blueprint to modernize the United Nations, Annan told more than 150 presidents, prime ministers and kings Wednesday that "a good start" had been made with the document.
But he also said leaders must "be frank with each other, and the peoples of the United Nations. We have not yet achieved the sweeping and fundamental reform that I and many others believe is required."
The summit began a week after investigators criticized alleged corruption and U.N. mismanagement of the oil-for-food program in Iraq, and on a day when more than 160 people died in attacks in Baghdad — a harsh reminder of the fight against terrorism that was a highlight of President Bush's speech.
A key goal of the summit is to take action to achieve U.N. Millennium Development Goals, a set of eight targets meant to reduce global poverty and disease.
The leader of the Netherlands challenged other rich nations to join the handful of countries that have committed to setting aside 0.7 percent of gross national product for overseas development aid. The United States strongly opposes the target.
Prime Minister Jan Peter Balkenende said there's only a slim chance as it is of meeting the millennium goals.
"The shortfalls are serious. Nothing less than an extra 50 to 60 billion (dollars) must be raised every year" to achieve millennium goals, he told the U.N. World Summit.
President Bush broadened the terrorism fight beyond the military arena, saying world leaders have "a solemn obligation" to stop terrorism in its early stages.