Israeli and Palestinian negotiators gave themselves about nine months to try to reach an agreement on ending their conflict of more than six decades in US-brokered peace talks.
The two sides held their first peace negotiations in nearly three years in Washington yesterday and today.
Flanked by the chief negotiators for both sides, US Secretary of State John Kerry said he was well aware of the doubts but described the initial talks as "constructive and positive."
Speaking after the meetings Mr Kerry said he believed peace was possible despite the obstacles.
The talks also included a session at the White House with President Barack Obama and Vice President Joe Biden, as well as talks between the two sides without US officials present.
The talks will go to a second round by the middle of August.
The resumption of negotiations is a rare moment of good news in the Middle East for the Obama administration, which has struggled to formulate a policy to try to end the civil war in Syria or to facilitate a democratic transition in Egypt.
Mr Kerry, who has prodded, coaxed and cajoled the two sides to resume negotiations in a flurry of visits to the Middle East during his less than six months in office, urged the Israelis and Palestinians to embrace "reasonable, principled compromise."
However, Palestinian President Mahmoud Abbas said yesterday that no Israeli settlers or border forces could remain in a future Palestinian state and that Palestinians deem illegal all Jewish settlement building within the land occupied in the 1967 Middle East war.
"In a final resolution, we would not see the presence of a single Israeli - civilian or soldier - on our lands," Mr Abbas said in a briefing to mostly Egyptian journalists.
The United States is seeking to broker an agreement on a two-state solution, in which Israel would exist peacefully alongside a new Palestinian state created in the West Bank and the Gaza Strip, lands occupied by the Israelis since the 1967 war.
The last direct negotiations collapsed in late 2010 over Israel's construction of Jewish settlements in the occupied territory.
The major "final-status" issues to be resolved include borders, the future of Jewish settlements in the West Bank, the fate of Palestinian refugees and the status of Jerusalem.