President Barack Obama has numerous world issues to address
By Paul Reynolds
World affairs correspondent, BBC News website
In his first foray into the annual meet-and-greet fest known as the UN General Assembly this week, US President Barack Obama will seek to make his impact on the major issues facing the world.
Having faced domestic arguments over his healthcare reforms and warnings over Afghanistan, the high expectations he has raised internationally will now be tested.
Mr Obama makes his first General Assembly speech on Wednesday. This should set the tone for his diplomacy over the next years.
This is his agenda.
Afghanistan is a huge problem for him, but it is principally a military one at the moment.
Iran is probably the most difficult diplomatic challenge facing him right now.
Because Iran is refusing to accept the demands from the Security Council for it to suspend the enrichment of uranium, the US and its allies are trying to rally support for increased sanctions, especially on Iran's oil and gas industry.
Mr Obama is trying to get Russia and China's support. He will have meetings with Chinese President Hu Jintao and Russian President Dmitry Medvedev at which this is expected to be discussed.
At the same time this week, Iran's President Mahmoud Ahmadinejad will be making his impact at the UN, probably refusing to stop enrichment but maybe saying, as the Iranians have been stating recently, that they do not intend to make a nuclear bomb.
Mr Obama, unlike President George W Bush, has placed the United States in the centre of the negotiations that are supposed to lead to a new post-Kyoto agreement in Copenhagen in December.
China's President Hu Jintao has to deal with his country's high pollution
He will speak at a special "high-level event" called by the UN Secretary General for heads of state and government.
Expect the president to lay out his aims for Copenhagen.
Crucial for the success of Copenhagen will be an understanding between the two largest polluters, the US and China.
Update: President Obama said that "the time we have to reverse this tide is running out." However his words will not be enough. US action also depends on the Senate where a climate change bill has yet to be passed. President Hu said that China would make "notable" reductions, seen as a significant advance in China's position.
The meeting with Mr Medvedev follows Mr Obama's decision to scrap the anti-missile system set to be based in Poland and the Czech Republic.
Although the US said it was designed against potential missiles from Iran, the Russians objected.
Now they are pleased. Will this new atmosphere lead to other diplomatic rapprochements?
The US and Russia are aiming for a new nuclear arms agreement by December. That might now be more far-reaching. A key issue is whether Russia will join new sanctions on Iran.
On Thursday Mr Obama will chair a special nuclear disarmament meeting by the Security Council, a first for a US president. It is a signal that he wants to make progress in advance of next year's review conference on the Nuclear Non-Proliferation Treaty.
He is reported to have ambitious plans, wanting far fewer nuclear weapons and perhaps no practical testing of warheads.
Also in New York there is a conference on the Test Ban Treaty, which has not yet been ratified by enough signatories to come into force. Mr Obama has pledged to try to get it ratified, but that decision is with the US Senate.
On Tuesday he holds separate talks and then a joint meeting with the Israeli Prime Minister Benjamin Netanyahu and the Palestinian Authority President Mahmoud Abbas.
Israeli settlements remain a hurdle for Middle East peace talks.
The White House is playing down expectations. Spokesman Robert Gibbs said: "We have no grand expectations out of one meeting."
Mr Obama's hope had been that by now conditions would have been set for direct Israeli-Palestinian talks, which this summit could have launched.
One of those conditions was an Israeli freeze on settlements. The terms of that freeze have not been agreed.
If the talks result in such an agreement, paving the way for direct negotiations, it will be judged a success.
Update: there was no such agreement. The president said nevertheless that "permanent status negotiations must begin and begin soon", implying that the freeze condition is being loosened.
G20 and the world economy
On Thursday the roadshow moves to Pittsburgh for a meeting of the G20 grouping of 19 industrial countries and the EU as a whole.
The major issue is whether governments should continue with stimulus packages to help them out of recession or whether they can now afford to ease off given that growth is returning to some economies.
They will also discuss reforms to the international financial institutions and the controversial problem of bankers' salaries and bonuses. The climate change negotiations will also be discussed.