After struggling mightily in a long exchange about his treatment of women, Donald J. Trump lashed out at Hillary Clinton and pledged that as president he would appoint a special prosecutor to pursue her.

Mr. Trump repeatedly interrupted Mrs. Clinton and taunted her as she sought to speak, saying several times: “You should be ashamed of yourself.” He accused her of covering for her husband’s alleged abuse of women, and of deliberately destroying evidence in a congressional probe of her emails.

Mrs. Clinton remained impassive through most of Mr. Trump’s extended attack, jabbing just once that Mr. Trump was eager to distract from “your campaign and the way it’s exploding.”

But it was Mrs. Clinton who delivered the most pointed denunciation of the evening, minutes before Mr. Trump’s tirade, calling him to account for his attacks on the Khan family, a Hispanic federal judge, President Obama and a newspaper reporter with a physical disability.

“He owes our country an apology,” Mrs. Clinton said. “And he needs to take responsibility for his actions and his words.”

  

Here are the rest of the highlights:

• Mrs. Clinton and Mr. Trump took the stage without shaking hands.

• Asked by an audience member about coarseness in the presidential race, Mrs. Clinton said it was “very important for us to make clear to our children that our country really is great because we’re good.” She reminded the crowd of her campaign slogan, “Stronger Together.” Mr. Trump, who before the debate appeared with women who have accused Bill Clinton of sexual assault, said he broadly agreed with Mrs. Clinton. “I began this campaign because I was so tired of seeing such foolish things happen to our country,” he said.

• Pressed on the 2005 recording which he seemed to boast of sexually assaulting women, Mr. Trump said he was “not proud of” the behavior, saying that he had apologized to his family and the American people. But he disputed that the recording amounted to bragging about sexual assault, calling his comments “locker room talk.” “I have great respect for women,” he said. “Nobody has more respect for women than I do,” he said, adding, “I was embarrassed by it.”

• Mrs. Clinton, responding to Mr. Trump’s remarks about the recording, said that while she disagreed with past Republican nominees, “I never questioned their fitness to serve.” “Donald Trump is different,” she said. She suggested that, despite Mr. Trump’s insistence that the recording did not reflect his character, “It’s clear to anyone who heard it that it represents exactly who he is.”

• Mr. Trump turned the discussion of his lewd remarks on Mr. Clinton, arguing that his “words” did not compare to Mr. Clinton’s history with women. “If you look at Bill Clinton, far worse,” Mr. Trump said. Mrs. Clinton did not address the attacks on her husband. “He gets to run his campaign any way he chooses,” she said. She paraphrased Michelle Obama’s speech at the Democratic convention to describe her approach: “When they go low, you go high.”

• Addressing Mrs. Clinton, Mr. Trump said that if he wins, “I am going to instruct my attorney general to get a special prosecutor to look into your situation,” citing her use of a private email server as secretary of state. Mrs. Clinton pointedly said she could not spend all of her time fact-checking Mr. Trump, advising viewers to go to her website to see his falsehoods. She said it was a good thing that Mr. Trump was not in charge of the laws in the country. “Because you’d be in jail,” Mr. Trump shot back. Cheers could be heard from the crowd.

• Mrs. Clinton said Mr. Trump was plainly trying to divert attention from his own campaign — “the way it’s exploding and the way Republicans are leaving you.” Mr. Trump asked a moderator, Mr. Cooper, why he had not spent more time discussing Mrs. Clinton’s private email server. (The other moderator, Martha Raddatz, had in fact brought it up moments earlier.) Mr. Trump was unmoved. “Nice, one on three,” he said, suggesting the moderators were teaming up on him.

• In response to a question on the Affordable Care Act, Mr. Trump called the program “a disaster.” “You know it, we all know it,” he said. Mrs. Clinton was asked about recent remarks from her husband, in which he appeared to criticize Mr. Obama’s signature legislative achievement. “He clarified what he meant,” Mrs. Clinton said. “If we were to start all over again, we might come up with a different system,” she said. “But we have an employer-based system.”

• Asked about how to stop Islamophobia, Mr. Trump said, “You’re right about Islamophobia and that’s a shame.” But he pivoted immediately to a discussion of what he called “radical Islamic terrorists,” citing the mass shooting in Orlando, Fla., and the attacks on the World Trade Center, among other atrocities. Mrs. Clinton suggested Mr. Trump’s statements throughout the campaign had been destructive. “We are not at war with Islam,” she said. “It plays into the hands of the terrorists to act as though we are.”

• Pushed repeatedly to answer whether his proposed ban on Muslim immigration still stood, Mr. Trump deflected, saying his plans amounted to “extreme vetting” and accusing a moderator, Ms. Raddatz, of favoring Mrs. Clinton. Mrs. Clinton said it was “important for us, as a policy” not to ban people based on religion. “How do you do that?” she asked. “We are a country founded on religious freedom and liberty.”

• After Mr. Trump said again that he opposed the war in Iraq, despite past public statements that contradict him, Mrs. Clinton reiterated that “we have it on tape” that Mr. Trump had not been against the war before it began. “It’s not been debunked,” he insisted, turning the issue back on Mrs. Clinton. “You voted for it and you shouldn’t have.”