The US presidential candidates squared off in a 90-minute debate at Hofstra University in New York, which was screened to tens of millions of television viewers in America and live-streamed across the world.
During a number of fiery exchanges, Mr Trump claimed he had a "better temperament" than Mrs Clinton and accused his Democrat rival of not having the "stamina" to be president.
"This is a man who has called women pigs, slobs and dogs," Mrs Clinton said.
Discussing the fight against Isis, Mr Trump attacked Mrs Clinton for revealing her plan to tackle the extremist group on her website.
"You're telling the enemy everything you want to do. No wonder you've been fighting Isis your entire life," the billionaire tycoon said.
The pair also addressed Mr Trump's refusal to release his tax returns, with Mr Trump insisting they were under a "routine audit".
But the Republican said he would release his tax returns against his lawyers' advice if Mrs Clinton releases 33,000 deleted emails from her private server.
His comments prompted cheers from sections of the audience, with moderator Lester Holt having to remind those watching to remain quiet.
Mrs Clinton said her use of a private email server was a "mistake" and claimed Mr Trump could be refusing to release his tax returns because he was not as rich or as charitable as he claimed.
She said: "I think probably he's not that enthusiastic having the rest of our country see what the real reasons are because it must be something really important, even terrible, he's trying to hide."
After Mr Trump claimed Mrs Clinton had described the Trans-Pacific Partnership trade deal as "the gold standard", she replied: "Donald I know you live in your own reality but that is not accurate".
Former President Bill Clinton and Melania Trump were among those in the audience for the debate.
Police on Long Island said about 2,000 protesters gathered outside the debate, including Hardhats for Hillary, socialists and activists calling for a living wage. Nassau County police said 24 people have been arrested on mostly disorderly conduct charges.
Mr Trump and Mrs Clinton will square off for two more debates next month, while their running mates, Tim Kaine and Mike Pence, are set to meet next week.
Mrs Clinton accused Mr Trump of starting his political career with a "racist lie" after he falsely suggested President Barack Obama was born outside of America.
Mr Trump said: "I got him to give the birth certificate so I'm satisfied with it."
Mrs Clinton said: "He has really started his political activity on this racist lie."
Mr Trump also repeatedly insisted that he opposed the Iraq War before the 2003 US and UK invasion, despite telling radio host Howard Stern in September 2002 that he supported it.
Explaining his 2002 comments, Mr Trump said: "I said very lightly, I don't know, maybe, who knows."
Discussing nuclear weapons, Mr Trump said he "would not do first strike" but he "can't take anything off the table".
Mrs Clinton said her Republican rival was too easily provoked to serve as commander in chief and could be quickly drawn into a war involving nuclear weapons.
"A man who can be provoked by a tweet should not have his fingers anywhere near the nuclear codes," she said.
Mr Trump replied: "That line's getting a little bit old."
Addressing Mr Trump's economic plans, Mrs Clinton said her Republican rival was promoting a "Trumped-up" version of trickle-down economics by focussing on tax cuts for the wealthy.
She called for increasing the federal minimum wage, spending more on infrastructure projects and guaranteeing equal pay for women.