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The Iran crisis
The annual gathering of political leaders at the United Nations is meant to be a chance to make the world a better place - tackle climate change, reduce poverty, improve women's rights.
But for this year's get-together one geopolitical headache hangs over proceedings - the bitter and increasingly militaristic standoff between America and Iran.
Over the summer there had been growing signs that Donald Trump is looking to change his "extreme pressure" campaign against Tehran into one focused on deal-making.
His hawkish national security adviser John Bolton, who once wrote a column headlined "To Stop Iran's Bomb, Bomb Iran", has been ousted from his role.
There was speculation the US president floated the idea of lowering sanctions on Iran and even, reportedly, flirted with a £15 billion bailout for the country.
Mr Trump has also repeatedly hinted that he would be willing to meet Hassan Rouhani, the Iranian president, at the UN this week.
But that was before the attack on two Saudi oil facilities which America - and now the UK - has blamed on Iran. While a meeting remains possible, the chances have been dropping by the day.
A Trump-Rouhani handshake would be historic, the first face-to-face talks between the countries' leaders in 40 years. But do not hold your breath.
India, Pakistan and the Kashmir stand-off
The United Nations confrontation between Narendra Modi and Imran Khan has already started. Pakistan showed it intended to make life difficult for the Indian PM from the beginning, by even refusing him access to their airspace to fly to the general assembly.
The source of enmity between the two, as so often, is Kashmir.
Pakistan has decried Mr Modi's ending of self rule for Indian-administered Kashmir and vowed to use the UN to highlight Delhi's alleged abuses. Mr Khan's address is seen as a golden opportunity to court international favour to Islamabad's position. Pakistan has so far failed to gain much diplomatic traction on the issue and Mr Khan's speech is likely to be forceful and combative.
India has in turn said it is prepared for Pakistan to “stoop low” at the gathering.
Yet amid the stand-off, both men will also be hoping for quality time with Donald Trump.
Mr Khan wants to continue rekindling relations with Pakistan's previously estranged allies in Washington and Mr Modi wants to ease US trade tensions. Mr Trump has for his part said he believes he can reconcile the nuclear-armed neighbours. Expect fireworks.
Boris Johnson, Leo Varadkar and the Irish border
Leo Varadkar, more than any other EU leader, holds the keys to the Brexit deal that Boris Johnson says he craves.
The Irish prime minister has, along with the European Commission, insisted on the need for the border backstop that Mr Johnson has pledged to tear up.
The rest of the EU has rallied behind Ireland and will not agree a replacement to the backstop without Dublin’s say-so.
Mr Varadkar, who has forged a close alliance with Michel Barnier is adamant any replacement must keep the border invisible and safeguard the Good Friday Agreement. Mr Johnson’s suggestion for an alternative fall short of that mark, according to Dublin and Brussels.
A no deal Brexit will hit Ireland harder than any other country except Britain. No surprise then that relations between Mr Varadkar and Mr Johnson are strained.
During the prime minister’s visit to Dublin on September 9, Mr Varadkar warned Mr Johnson he faced a “Herculean” task in delivering Brexit and a free trade deal with the EU.
In a barbed remark, the Taoiseach suggested he would be Mr Johnson’s Athena. Athena knocked out a rampaging Hercules, preventing him from causing more carnage, after he murdered his wife and children.
On September 20, Mr Varadkar was handed a vial of holy water by a priest before “for use at his meeting with Boris Johnson”.
“Do I just throw it over him?,” Mr Varadkar joked.
Emmanuel Macron vs Jair Bolsonaro
Emmanuel Macron intends to come to New York as the voice of reason. He aims, as last year, to present France as the last bastion of sensible multilateralism and cool-headed authority.
But if there is one man hoping to burst his bubble, it’s Jair Bolsonaro, the combative president of Brazil, who took office on January 1.
The two men fell out spectacularly this summer, after Mr Macron suggested that the fires in the Amazon – the worst in a decade – were a global problem, that needed international help. Mr Bolsonaro, a far-Right populist who is sceptical of climate change and wants to open up more of the Amazon to logging, hit back, accusing Mr Macron of having “a colonial mindset”. He then went on to mock the looks of Mr Macron’s wife. Mr Macron called him “extremely rude” and adding: “I hope that they will very quickly have a president who behaves properly.”
Brazil, by tradition, gets to speak first at the UN General Assembly – meaning he can land the first punch. Expect him to hit hard, attacking those who seek to interfere in what he sees as Brazil’s internal affairs. Expect Mr Macron to be asked about it after, too – and for him to address the issue in his own speech, later on Tuesday.
Both sides are certainly fired up. Mr Bolsonaro’s tourism advisor – a mixed martial arts “legend” - Renzo Gracie, accused Mr Macron of “talking rubbish” about his country and challenged him to a fight.
“Come over here you’ll be caught by the neck, that chicken neck,” he said. “You don’t fool me.”