Statesmen and diplomats are attending one of the largest global summits in history at the United Nations headquarters in New York.
The UN building is protected by an extensive security operation
About 150 world leaders are at the 2005 World Summit, which marks 60 years since the founding of the UN.
The UN chief urged them to implement UN reforms and to take bolder steps to fight poverty and promote security.
Kofi Annan said the member states' deal on a reform package was a good start, but differences had prevented progress.
And President George W Bush said the US was committed to helping overcome poverty. "We have a moral obligation to help others," he said.
Mr Bush said terrorism, fed by anger and despair, easily spread across the world.
"We have not yet achieved the sweeping and fundamental reform that I and many others believe is required," Kofi Annan told the assembled leaders.
The secretary general said the biggest failing was in the areas of nuclear non-proliferation and disarmament, where "posturing" had got in the way of results.
He also expressed concern about the lack of agreement over Security Council reform.
The summit was preceded by bitter negotiations and an eventual compromise deal on plans for UN reform and proposals to tackle global poverty.
Plans to include disarmament and non-proliferation in the draft document were dropped during negotiations.
Mr Annan's long-held hope for major reform of the 15-member Council was deferred for later discussion.
Nevertheless, a 35-page draft declaration was adopted unanimously by all 191 UN members on Tuesday night.
Dozens of kings, queens, presidents and prime ministers will have the opportunity to speak to the General Assembly during the three-day summit.
The BBC's Bridget Kendall, in New York, says that ostensibly the summit is a chance to endorse at the highest level bold plans to make the UN more effective and able to tackle today's numerous global challenges.
But there were major disagreements over key issues during negotiations on a draft declaration.
Outgoing General Assembly President Jean Ping, of Gabon, presented delegates with a compromise text:
- The document does not offer an international definition of terrorism, although hope remains for an agreement during the summit
- Plans to reform the UN's much-criticised Human Rights Commission were also deferred to the General Assembly
- A commitment to break down trade barriers was substantially weakened
- Diplomats agreed on the creation of a peace-building commission to help nations emerging from war and on an obligation to intervene when civilians face genocide and war crimes
- A section on development voices support for the UN's Millennium Development Goals, a long-term strategy for eliminating world poverty.
Mr Annan was critical of the compromise text, but he said he hoped world leaders would take up the issues up at the summit.
The document, hastily pushed through at the last minute, falls well short of the original sweeping vision, our correspondent says.
Mr Ping described the document and the UN itself as a reflection of the international community.
He said that there was not the political will to adopt a 21st Century "Marshall Plan" approach to alleviating poverty.
A spokesman for the UK-based aid agency Oxfam criticised the compromise deal and accused several countries of undermining the plan.