Trump, a Republican, has described the 71-year-old United Nations as weak and incompetent and threatened to pull out of a global deal to combat climate change - a cornerstone of the legacy of UN chief Ban Ki-moon, who steps down at the end of 2016 after serving two five-year terms as secretary-general.
"The United Nations is not a friend of democracy, it's not a friend to freedom, it's not a friend even to the United States of America," Trump said during a speech in March to the American Israel Public Affairs Committee.
The United States is a veto-wielding member of the 15-member UN Security Council and the largest financial contributor to the United Nations. Washington currently owes some $1.1 billion, the United Nations said. Republicans have long been reluctant to pay dues, accusing the world body of waste and bias.
Ban said on Wednesday he hoped that the Trump administration would "strengthen the bonds of international cooperation."
"People everywhere look to the United States to use its remarkable power to help lift humanity up and to work for the common good," Ban told reporters.
Topping the United Nations agenda are conflicts in Syria, Yemen, South Sudan, Iraq and elsewhere, and a refugee and migrant crisis that saw a record 65.3 million people uprooted worldwide last year.
A senior Security Council diplomat, speaking on condition of anonymity, said Trump's foreign policy had so far "not been very coherent" and his victory did not bode well for the future effectiveness of the council.
Several UN diplomats noted a lack clarity from Trump on foreign policy.
"We have not really heard a very clear indication of where he is going," Iraq's UN Ambassador Mohamed Ali Alhakim told reporters on Wednesday, adding that he believed the United Nations would remain relevant.
"If there is any change in the policy we will then determine later on what we're going to do with that," he said. "Others were saying this is similar to Brexit, but this is what the Americans selected."
Yemen's UN Ambassador Khaled Alyemany said he hoped the "new president will contribute positively to resolve the crises in the Middle East," acknowledging that Trump had not laid out a foreign policy plan for people to see.
The United Nations High Commissioner for Human Rights, Zeid Ra'ad al Hussein, last month warned that the world would be in danger if Trump was elected president. Zeid would continue to call out any Trump policies or practices that undermine or violate human rights, his spokesman said.
A western diplomat, speaking on condition of anonymity, summed up how Trump's victory could impact the United Nations: "I think there will be a long hangover."
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