Former Secretary of State Hillary Clinton remains a frontrunner.
Her main rival, Senator Bernie Sanders, apologised for his campaign's accessing of voter information compiled by Ms Clinton's team.
"Not only do I apologise to Secretary Clinton, I want to apologise to our supporters," he said.
Former Maryland Governor Martin O'Malley is also taking part in the debate in Manchester, New Hampshire.
Gun control and the fear of terror attacks in the United States were tackled early in the debate.
"Guns in and of themselves will not make America safer," said Hillary Clinton.
"Arming more peopleâ€¦ is not the appropriate response to terrorism."
Ms Clinton said that Republican rhetoric - especially from Donald Trump - was "fanning the flames of radicalisation".
But Mr O'Malley accused of two rivals of "flip-flopping" in Washington over gun ownership restrictions.
For his part, Mr Sanders said US foreign policy needed to focus on destroying the so-called Islamic State (IS or Isis) group, not removing Bashar al-Assad from power in Syria.
"It is not Assad that is attacking the United States," he said.
"First priority: destroy Isis," he said.
Mr Sanders also criticised Qatar, saying that "instead of spending $200bn on the World Cup, maybe they should pay attention to Isis on their doorstep".
Bernie Sanders said that on two occasions there were computer breaches - which he said were the fault of the software vendor - in which his campaign could see proprietary data from Hillary Clinton's campaign and that they alerted party officials "quietly".
He said his campaign "didn't run to the media and make a big deal about it".
He said that the most recent breech involved inappropriate behaviour by one of his staff members, adding that person had now been dismissed.
He said that the Democratic Party's decision to temporarily suspend his campaign's access to the strategically crucial database was "an egregious act".
The Sanders campaign on Friday filed a lawsuit against the Democratic National Committee in a federal court to restore its access to the voter data.
Strategically important information on voters is contained in the database, which campaigns use to decide strategy.
That data takes on a crucial role as campaigns prepare for early primary voting in just over a month's time.
Forbidding the Sanders campaign from accessing the database will significantly hamper its ability to reach out to new Democratic voters, says the BBC's Anthony Zurcher in Washington.
The DNC maintains the database and charges campaigns for access.
Primary voting begins in early February.

She beat the 21st seed 4-6 6-4 8-6 in the fourth round, in a match that lasted three hours and four minutes.
"It was mentally, emotionally and physically draining," Konta said.
At 24 years old, Konta is ranked 47th in the world.
Her next opponent will either be 15th seed Madison Keys, or China's Shuai Zhang.
The 'Grand Slam' tournaments are the four most important annual tennis competitions. Players travel from around the world to take part in them.
They include the Australian Open, US Open, Roland Garros, and Wimbledon.
The last British woman to play in the quarter-finals of a Grand Slam was Jo Durie, at Wimbledon in 1984.
Candidates seeking the Democratic Party's nomination for the US presidency have begun their last debate of the year in New Hampshire.
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Johanna Konta has become the first British woman to reach a Grand Slam quarter-final for 31 years, after beating Ekaterina Makarova at the Australian Open.