At first, Trump did not name the North Korean leader, using a nickname instead.
Trump said, "No nation on earth has an interest in seeing this band of criminals arm itself with nuclear weapons and missiles. The United States has great strength and patience, but if it is forced to defend itself or its allies, we will have no choice but to totally destroy North Korea. Rocket Man is on a suicide mission for himself and for his regime. The United States is ready, willing and able, but hopefully this will not be necessary."
He also said it was "an outrage" that some nations continue to trade with North Korea. At other points, he referred to Kim's regime as "depraved" and "twisted."
The North Korean ambassador to the United Nations, Ja Song Nam, was photographed at the General Assembly this morning, but he left before Trump entered the room. The seats assigned to North Korea, which are in the front row, were empty during Trump's speech. The only individual in the area was an unidentified man who appeared to be taking notes.
Slamming the Syrian regime
Trump then turned his attention to Syrian President Bashar al-Assad and Venezuelan President Nicolas Maduro.
Trump defended U.S. strikes on an air force base in Syria this year after a chemical attack by government forces on a rebel-held area in the country's northwest.
"The actions of the criminal regime of Bashar al-Assad — including the use of chemical weapons against his own people, even innocent children — shock the conscience of every decent person. No society can be safe if banned chemical weapons are allowed to spread," Trump said.
The 'very real crisis' in Venezuela
On Venezuela, Trump said the goal is for the country's people to "regain their freedom, recover their country and restore their democracy."
"We are prepared to take further action if the government of Venezuela persists" on its authoritarian path, he said. "The problem in Venezuela is not that socialism has been poorly implemented but that socialism has been faithfully implemented," he said.
He continued, "Major portions of the world are in conflict, and some, in fact, are going to hell. But the powerful people in this room, under the guidance and auspices of the United Nations, can solve many of these vicious and complex problems."
Criticizing Iran and the nuclear deal
Trump confronted Iran's destabilizing behavior in the Middle East, leaving open the possibility that the U.S. will unilaterally pull out of the multinational nuclear deal with the country — which he deemed an "embarrassment" — and made an appeal to the Iranian people to stand against their government and seize their future.
Trump said its government "must stop supporting terrorists, begin serving its own people and respect the sovereign rights of its neighbors."
"The Iranian government masks a corrupt dictatorship behind the false guise of a democracy. It has turned a wealthy country, with a rich history and culture, into an economically depleted rogue state whose chief exports are violence, bloodshed and chaos," he said.
Talking about leading by example
Earlier in his speech, Trump touted his "America first" views and the importance of national sovereignty.
"As president of the United States, I will always put America first, just like you, as the leaders of your countries, as always and should always put your countries first," he said.
"We can no longer be taken advantage of or enter into a deal," he went on.
"In America we do not seek to impose our way of life on anyone but rather let it shine as an example for everyone to watch," he said.
Trump said America stands for "principled realism rooted in shared goals [and] values."
"We must work together and confront together those who threaten us with chaos," he said.