Israeli and Palestinian negotiators have held talks in Jerusalem on what are seen as the most intractable issues in the peace process.
There have been a series of meetings since the Annapolis talks
The meeting lasted two hours and were described as positive by both sides.
The talks were led by Israeli Foreign Minister Tzipi Livni and former Palestinian Prime Minister Ahmed Qurei.
The core issues include the status of Jerusalem, the borders of a Palestinian state, settlements in the West Bank, refugees, security and water resources.
Mr Qurei said the talks had been positive and the core issues were discussed in general terms, but warned the path ahead would be difficult.
An Israeli foreign ministry official said that further discussions would be intensive.
The AFP news agency quoted an Israeli official as saying the atmosphere in the talks was good and constructive.
Ms Livni said that details of the talks were unlikely to be revealed.
"Past experience has shown that when talks are held in the limelight they lead to the radicalisation of the positions and to the distortion of the things said behind closed doors; to a rise in expectation and to disappointment that eventually leads to violence," she said.
Also on Monday, Israeli Prime Minister Ehud Olmert warned a committee of the Israeli parliament of the difficulty of reaching a deal.
"I'm not sure we can reach an agreement and I'm not sure we can reach its implementation," he said according to AFP.
"But I will be committing a sin to my duty if I didn't try," he went on.
Prospects for a comprehensive settlement were given fresh impetus by the US peace conference at Annapolis, Maryland, in November, after which Mr Abbas and Prime Minister Olmert agreed to restart talks after seven years.
US President George W Bush, who visited Israel and the West Bank last week, has said Israel and the Palestinians could sign a peace treaty within a year.
"I believe it's going to happen, that there will be a signed peace treaty by the time I leave office," he told reporters in Ramallah on Thursday.
Offer to Hamas
In a speech on Sunday, Mr Abbas said he had warned the US president that the Palestinians could not move ahead in the negotiations while Jewish settlements continued to be built in the West Bank.
"We can't have negotiations while they are building houses all over," he added.
The Palestinian leader also said he was willing to restart talks with the Islamist movement, Hamas, if it relinquished control of the Gaza Strip.
Hamas ousted Mr Abbas's Fatah movement from the coastal territory in June following a week of violent clashes.
"I am ready to negotiate with Hamas even if the United States does not accept it," he said. "Such a dialogue is very important."
A Hamas statement denounced Mr Abbas's speech as "full of lies and fabrications".