■ Mrs. Clinton pressed Mr. Trump’s repeated claim, contradicted by public statements he has made in the past, that he had opposed the Iraq war from the beginning. “Donald supported the invasion of Iraq,” she said. “That is absolutely proved over and over again.” “Wrong,” Mr. Trump interjected. “Wrong.”

■ Mr. Trump insisted again that he had opposed the Iraq war, calling any suggestion otherwise “a mainstream media nonsense.” Lester Holt, the moderator, said, “The record shows otherwise.” Mr. Trump went on to appraise his own temperament, calling it “my strongest asset, maybe by far,” before attacking Mrs. Clinton’s. She smiled. “Woo! O.K.,” she said, beginning her response.

■ Mrs. Clinton hit Mr. Trump for his record of praise for President Vladimir V. Putin of Russia, lamenting that Mr. Trump had “publicly invited Putin to hack into” American communications, as Mr. Trump shook his head. Mr. Trump said he did not think “anybody knows” if Russia was behind the recent hacks of Democratic organizations, wondering aloud if it might instead be China or a single hacker who “weighs 400 pounds” and sits at home.

■ After Mrs. Clinton suggested she was worried about Mr. Trump getting his hands on the nuclear codes, Mr. Trump replied, “That line is getting a little bit old.” Mrs. Clinton mocked Mr. Trump’s “secret plan,” as she called it, to fight the Islamic State, saying he had no true strategy. She also sought to reassure American allies that the country would honor its international commitments, saying some of Mr. Trump’s comments during the campaign had startled them.

■ Asked about his propagation of the conspiracy theory raising doubts about Mr. Obama’s birth, Mr. Trump tried to blame Mrs. Clinton, suggesting falsely that she, too, had questioned the president’s birthplace. “She failed to get the birth certificate,” he said. “When I got involved, I didn’t fail.” Mrs. Clinton said Mr. Trump had “really started his political activity based on this racist lie that our first black president was not an American citizen,” calling his efforts “very hurtful.”