In opening some four hours of talks in Washington, US Secretary of State Hillary Clinton said she understood the "suspicion and skepticism" leading up to the meetings.
"I know the decision to sit at this table was not easy," Clinton told Abbas and Netanyahu in a chandeliered room at the State Department, joined by their delegations. "Thank you for your courage and commitment."
After Wednesday's weighty symbolism and lofty rhetoric, culminating in a White House dinner with President Barack Obama, Netanyahu and Abbas finally got down to the business end of proceedings, presenting their opening demands.
"We expect you to be prepared to recognize Israel as the nation state of the Jewish people," Netanyahu told Abbas, as the two sat on either side of Clinton with their national flags behind them.
Such a move would be politically difficult for Abbas as it could undermine the right-of-return claims of Palestinian refugees who left or fled Israel when it was created in 1948.
In the wake of two Palestinian militant attacks on settlers in the occupied West Bank, Netanyahu renewed references he had made at the White House on Wednesday to protecting Israel's security.
"A real peace must take into account the genuine security needs of Israel," he said.
Abbas appeared conciliatory, saying: "We consider security as essential and vital both for us and for you, and we will not accept that anyone commits any act that would harm your security or ours."
The Palestinian leader said investigations into the shootings that killed four Israeli settlers on Tuesday were progressing.
But he also stuck to his demands on settlements.
"We call on the Israeli government to move forward with its commitment to end all settlement activity and completely lift the embargo over the Gaza Strip," Abbas said.
Israel tightly controls access and egress from the Gaza Strip, which is run by the militant Hamas group, a rival of Abbas's Palestinian Authority and a fierce opponent of the peace talks.
Before going behind closed doors to begin tackling the core issues that have bedeviled past peace attempts, the two leaders poignantly shook hands, and Abbas appeared to give Netanyahu a thumbs up.
US Middle East envoy George Mitchell described a first 90-minute meeting between him Clinton, Abbas and Netanyahu as "productive" and said the parties agreed the core issues could be settled within a year.
Netanyahu and Abbas then held a one-on-one meeting with no note-takers or translators which officials said lasted 93 minutes.
The Palestinians said Abbas had reiterated face-to-face with Netanyahu his demand that settlement construction must cease for the talks to continue.
A final meeting with both leaders, Clinton and Mitchell lasted another 18 minutes.
Abbas and Netanyahu agreed to resume talks in two weeks and planned to meet every two weeks thereafter. Mitchell and Clinton announced their intention to be present at the future talks.
A Palestinian official said the talks on September 14-15 would take place in Sharm el-Sheikh, where negotiations have taken place in the past.
Mitchell said: "Our goal is to resolve all of the... core issues within one year. And the parties themselves have suggested and agreed that the logical way to proceed to tackle them is to try to reach a framework agreement first."
Such an agreement is less detailed than a "full-fledged treaty" but more detailed than a declaration of principles.
Abbas had previously refused to enter direct negotiations without a full halt to Israeli settlement activity, but yielded under pressure from Obama.
The Palestinian leader is still warning that a renewal of settlement activities after September 26, when a 10-month partial moratorium expires, would end the negotiations.
The last direct peace negotiations ended in December 2008 when Israeli forces invaded Gaza to halt Hamas rocket fire on Israel.
-------------------------------------------------------------------------