Trump says he has great respect for the leader of China but the USA trade imbalance with Beijing "cannot be tolerated".
These are the highlights of Donald Trump's speech.
Iran's behavior will also be a major focus Wednesday when Trump leads a meeting of the U.N. Security Council.
While relations with Kim have improved dramatically, leaders attending the annual assembly expect this year to hear Trump espouse a hard line on a different USA adversary, Iran's Hassan Rouhani.
"They do not respect their neighbours or borders or the sovereign rights of nations".
"We can not allow a regime that chants "Death to America" and that threatens Israel with annihilation, to possess the means to deliver a nuclear warhead to any city on Earth", he said. Rod Rosenstein was reported last week to have floated the idea of secretly recording the president past year and to have raised the idea of using the 25th Amendment to remove him from office.
Iran: "Iran's leaders sow chaos, death and destruction", he said.
Speaking in triumphal terms, Trump approached the address as an annual report to the world on his country's progress since his inauguration.
"The missiles and rockets are no longer flying in every direction", he said. He received a stay of punishment at least until Thursday, when he is to meet with Trump at the White House.
Despite Trump's praise, the administration has said Pyongyang is slow-walking toward fulfilling the goals set in Singapore.
"That is why America will always choose independence and cooperation over global governments, control and domination".More news: GOP pushing forward for Kavanaugh, accuser wants fairness
Donald Trump's attack against the International Criminal Court, an organization the United Nations indirectly works with, although not unexpected, was clearly a statement of not backing down.
Even President Donald Trump wasn't expecting the reaction he got at the United Nations when boasting about his administration's accomplishments.
He added, "In less than two years, my administration has accomplished more than nearly any administration in the history of our country". His boast elicited laughter from the scores of heads of state and delegates in the audience for the speech.
US President Donald Trump said yesterday he expects to hold a second summit soon with Kim Jong-un, lauding the North Korean strongman at the United Nations as "terrific" one year after he eviscerated him from the same platform.
Over the summer, Mr. Trump had said he would meet with Rouhani without preconditions to negotiate a new deal, an offer reiterated on Sunday by U.S. Secretary of State Mike Pompeo and extended to Iran's supreme leader, Ayatollah Khamenei.
Trump, who begins his political rallies with boasts about his economic record in less than two years in office, used the same rhetoric before the crowd of world leaders and diplomats, telling them he had accomplished more than nearly any previous US president.
Some Iranian insiders have said any talks between Rouhani and Trump would effectively kill the existing nuclear accord.
Trump said he will meet with Rosenstein on Thursday upon his return to Washington.
India is also seeking a waiver from US sanctions to buy Iranian oil and might create its own alternative payment system if the request is not granted.
Trump says he looks forward to having a "great relationship" with Iran but "Iran has to change its tune".
Mr. Trump compared USA relations with Iran to what he called improved ties with North Korean leader Kim Jong Un, praising him for halting nuclear and missile tests and returning some US remains from the 1950s Korean War.