The Democrat and Republican rivals 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 her of not having the “stamina” to be president.
Mrs Clinton said Mr Trump had started his political career with a “racist lie” after he falsely suggested President Barack Obama was born outside of America.
“He has a long record of engaging in racist behaviour,” she said.
Mrs Clinton also criticised the billionaire tycoon’s comments about women, adding: “This is a man who has called women pigs, slobs and dogs.”
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,” he said.
The candidates also addressed Mr Trump’s refusal to release his tax returns, with the Republican insisting he would publish them against his lawyers’ advice if Mrs Clinton releases the 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 the Republican 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.
Discussing nuclear weapons, Mr Trump said he “would not do first strike” but he “can’t take anything off the table”.
Other notable exchanges during included Mrs Clinton telling Mr Trump “you live in your own reality”, before adding “By the end of this evening I think I’ll be blamed for everything that has ever happened.”
Mr Trump prompted laughter from the audience when he replied: “Why not?”
Former President Bill Clinton and Melania Trump were among those in the audience at the event, which was expected to the most watched debate in TV history.
HERE ARE 10 KEY POINTS FROM THE DEBATE
• Jobs and the economy:
The candidates kicked off with different approaches to the first question on Americans’ wealth and work, with Mrs Clinton looking to future policies while Mr Trump targeted the Democrats’ record in government. Mrs Clinton noted that it was her granddaughter Charlotte’s second birthday as she pledged to achieve gender parity in the workplace and increase taxes for the wealthy.
Mr Trump said the US had lost jobs to Mexico and industry to China and promised to revive the US economy by lowering taxes, cutting regulation and renegotiating trade deals. He repeatedly attacked his rival’s husband, former US president Bill Clinton, for the North American Free Trade Agreement which was approved under his administration in the 1990s.
Mrs Clinton used the subject of family income to launch a personal attack on Mr Trump, contrasting his charmed start in life with her own.
• Emails and tax returns:
These were two subjects on which both candidates could expect to be attacked. Mrs Clinton admitted she had made a mistake by using a private email system during her tenure as US secretary of state. Mr Trump said he would only disclose his tax affairs if his rival releases the “33,000 emails” deleted from her private server. However Mrs Clinton accused Mr Trump of hiding “something terrible”, suggesting he may not be as wealthy as he claims, pay a low rate of tax or give smaller charitable donations than he has claimed.
• Climate change:
Mr Trump denied saying climate change was false despite having tweeted in January 2014: “Snowing in Texas and Louisiana, record setting freezing temperatures throughout the country and beyond. Global warming is an expensive hoax!”
He also claimed in 2012 that the concept of global warming was “created by and for the Chinese in order to make US manufacturing non-competitive”.
The candidates were asked how they would heal race relations in the wake of police shootings that have provoked outrage across the US. Mrs Clinton said gun violence is the leading cause of death among young African-American men and tackling the “plague of gun violence” is critical.
The pair clashed on “stop and frisk” laws, with Mr Trump claiming a judge’s ruling that the practice was unconstitutional was wrong. He said: “Right now our police are afraid of doing anything” and that if you walk down the streets in places like Chicago, “you get shot”. A theme throughout his campaign has been to “make America safe again”.
• “Birther” controversy:
Mr Trump recently dropped a conspiracy theory that Barack Obama was not born in the United States and was therefore ineligible to serve as president. The retreat came after years of campaigning for Mr Obama to release his birth certificate. In the debate Mr Trump blamed Mrs Clinton for starting the false claim that Mr Obama was born in Kenya and said that he was instrumental in achieving the certificate’s release. Mrs Clinton accused her rival of spreading a “racist lie” that our “first black president” was not an American citizen.
During an exchange over her ability to cope with the physical demands of the job, Mrs Clinton referred to an episode between Mr Trump and US chat show host Rosie O’Donnell. “This is a man who has called women pigs, slobs and dogs,” the Democrat said.
• Gun laws:
On a rare point of agreement between the candidates, Mr Trump said he supported Mrs Clinton’s view that people on watch lists or no-fly lists should have restrictions on their ability to buy guns.
“We have to look very strongly at no-fly lists and watch lists,” the tycoon said.
• Defence and cyber security:
After batting off questions over his temperament, Mr Trump said he would not use America’s nuclear arsenal unless the US was struck with warheads first. However he also said he “can’t take anything off the table” and referred to issues with Iran and North Korea.
The candidates were quizzed on their views on cyber security, with a focus on Russia. Mrs Clinton said the US “is not going to sit idly by” and let hostile nations attempt to hack public or private information. Mr Trump’s past praise of Russian President Vladimir Putin left him open to attack. He simply replied “wrong” when the comments were raised.
Mr Trump was quick to bring national security into the debate even when the question was on a separate issue. He first accused Mrs Clinton of leaking tactics to Islamic State on her campaign website and criticised the Obama administration for the ascendancy of the terror group. The Democratic candidate said she is hopeful that IS will be eradicated by the end of the year - with taking out its leaders her top priority in office.
In the wake of several deadly terror-related incidents in the US, Mrs Clinton said her rival had “consistently insulted Muslims abroad, Muslims at home”, while people from within the Muslim community would be key to fighting terror.
• Fitness to serve:
Both candidates appeared fit and well during the debate - however, sniffling by Mr Trump set chins wagging on Twitter. After Mrs Clinton revealed earlier this month that she had been diagnosed with pneumonia, Mr Trump questioned whether she had the stamina to be president. “You have so many different things you have to be able to do and I don’t believe Hillary has the stamina,” he said. Mrs Clinton said Mr Trump should not talk about stamina until he has tried out the busy schedule she kept up as secretary of state.