The high-stakes showdown - the first of three presidential debates - comes as both candidates are viewed negatively by large numbers of Americans.
Democrat Ms Clinton is facing questions from the electorate about her trustworthiness, and Republican Mr Trump is struggling to convince many voters that he has the temperament and policy depth to be president.
In an occasionally rambunctious encounter, Hillary Clinton and Donald Trump interrupted each other repeatedly during the first US presidential debate.
Moderator Lester Holt struggled to rein in the candidates, with discussions about tax policy suddenly shifting to the fight against Islamic State.
Mr Trump accused Ms Clinton of giving away information to the enemy by revealing on her website how she planned to defeat the group.
Donald Trump claims, "She's going to raise taxes $1.3 trillion... "
"Mr Trump - "
"... and look at her website. It's no different than this. She's telling us how to fight ISIS. Just go to her website. She tells you how to fight ISIS on her website. I don't think General Douglas MacArthur would like that too much."
Moderator Lester Holt again tried to draw Mr Trump away.
"The next segment we're continuing the subject of -"
Ms Clinton then chimed in: "At least I have a plan to fight ISIS."
"No, no, you're telling the enemy everything you want to do."
"No, we're not."
"You're telling the enemy everything you want to do! No wonder you've been fighting ISIS your entire adult life."
"That's - please, fact-checkers, get to work."
As the 90-minute debate opened Ms Clinton raised the issue of leadership, a matter central to the campaign so far.
"We are going to have a debate where we are talking about the important issues facing our country. You have to judge us: who can shoulder the immense awesome responsibilities of the Presidency? Who can put into action the plans that will make your life better? I hope that I will be able to earn your vote on November 8th."
Mr Trump told the audience that he is far best suited to be the next President.
"I think my strongest asset, maybe by far, is my temperament. I have a winning temperament. I know how to win. The other day, behind the blue screen, I don't know who you were talking to, Secretary Clinton, but you were totally out of control. I said, 'There's a person with a temperament that's got a problem.'"
Donald Trump has criticised Ms Clinton for her trade policies, saying the government has to stop jobs from being stolen from the US.
Mr Trump says the US needs to re-negotiate trade deals.
He says jobs are disappearing from the US and going to countries like Mexico and China - and that has to stop.
"Our jobs are fleeing the country. They are going to Mexico. They are going to many other countries. You look at what China is doing to our country in terms of making our product. They are devaluing their currency and there is nobody in our Government to fight them. We have a very good fight and we have a winning fight. Because they are using our country as a piggy bank to rebuild China and many other countries are doing the same thing. We are losing our good jobs, so many of them."
Ms Clinton hit back, accusing Mr Trump's economic platform of amounting to the "most extreme" package of tax cuts for the wealthy in US history.
"And the kind of plan that Donald has put forth would be trickle-down economics all over again. It would be the most extreme version, the biggest tax cuts for the top per cent of the people in this country that we have ever had. I call it Trumped-up trickle-down. That is exactly what it would be. That is not how we grow the economy."
Mr Trump has repeatedly said in a Trump Presidency, US Foreign policy will be singularly focused on Making America Safe Again by destroying radical Islamic terrorist groups.
He's also pledged to end the nuclear deal with Iran, charge US allies for any assistance given to them and has been highly critical of the nation's defence policy.
Ms Clinton says his words are dangerous.
"Words matter. Words matter when you run for president and they really matter when you are president. And I want to reassure our allies in Japan and South Korea and elsewhere, that we have mutual defence treaties and we will honour them. It is essential that America's word be good and so I know that this campaign has caused some questioning and some worries on the part of many leaders across the globe."
Mr Trump says he supports all the US allies but it isn't up to America to help keep international peace for free.
"I want to help all of our allies but we are losing billions and billions of dollars. We cannot be the policeman of the world. We cannot protect countries all over the world with them not paying us what we need."
Both Mr Trump and Mrs Clinton hoped to use the debate to erase lingering voter doubts and address campaign-trail weaknesses.
Some issues were barely mentioned during the intense debate.
Illegal immigration and Mr Trump's promises of a border wall were not part of this conversation.
And while Ms Clinton took some questions on her private email server, she was not questioned about her family's foundation or her "supposed" struggle with trustworthiness.
There are two more debates to come before November 8, when electors cast their votes.
