Israeli Prime Minister Benjamin Netanyahu hailed Nobel Peace Prize winner Shimon Peres as a "great man of the world" and promised there would be peace as he eulogised him at his funeral.
US President Barack Obama and other world leaders gathered for the burial in Jerusalem's Mount Herzl cemetery, two days after Mr Peres, a former president and premier, died at the age of 93.
Palestinian President Mahmoud Abbas and Mr Netanyahu shook hands and exchanged brief words earlier.
The Government was represented by Minister for Foreign Affairs Charlie Flanagan.
In Jerusalem to pay my repects & those of the Irish people to the family of Shimon Peres & to sympathise with the people of Israel, pic.twitter.com/ifGtW9UYd2
— Charlie Flanagan (@CharlieFlanagan) September 29, 2016
"Shimon lived a life of purpose," Mr Netanyahu told those gathered at Jerusalem's Mount Herzl cemetery of the man who was once his political rival.
"He soared to incredible heights. He was a great man of Israel; he was a great man of the world.
"Israel grieves for him, the world grieves for him, but we find hope in his legacy, as does the world."
He recalled a late-night discussion between them in which they debated Israel's future.
"He said that peace was true security - if there is peace, there will be security," Mr Netanyahu said of the former prime minister and president.
We need your consent to load this rte-player contentWe use rte-player to manage extra content that can set cookies on your device and collect data about your activity. Please review their details and accept them to load the content.Manage Preferences
"I told him, 'Shimon, in the Middle East, security is an essential condition for peace, and for the establishment of peace' ... My friends, you know what surprising conclusion I came to? We were both right."
Later, Mr Netanyahu said "there will be peace, Shimon, dear man, exceptional leader. I tell you that from the bottom of my heart."
Mr Obama said Mr Peres reminded him of "giants of the 20th century" such as Nelson Mandela.
"In many ways, he reminded me of some other giants of the 20th century that I've had the honour to meet," Mr Obama said.
He also said that the Palestinian president's presence at the funeral was a reminder of the "unfinished business of peace".
Earlier while welcoming Mr Abbas, Mr Netanyahu said of his attendance: "It's something that I appreciate very much on behalf of our people and on behalf of us."
"Long time, long time," Mr Abbas said to Mr Netanyahu and the prime minister's wife Sara, after shaking his hand before the start of the state ceremony.
Mr Abbas's rare visit to the city, a short drive through Israeli military checkpoints from Ramallah, in the occupied West Bank, seemed unlikely to yield anything more than handshakes with Israeli Prime Minister Benjamin Netanyahu and Mr Obama.
Mr Peres jointly won a Nobel Prize for his peace efforts with the Palestinians in the 1990s.
Israeli-Palestinian negotiations have been frozen since 2014 and Mr Netanyahu and Mr Abbas, deeply divided over Jewish settlement on land Palestinians seek for a state and other issues, have not held face-to-face talks since 2010.
Mr Obama was to also deliver a eulogy, in what could be an opportunity for the US president to encourage Israelis and Palestinians to revive peacemaking.
US officials have held open the possibility of Mr Obama making another formal effort to get peace negotiations back on the agenda before he leaves office in January, possibly via a UN Security Council resolution.
Obituary: Shimon Peres, Israel's elder statesman
With time short between the end of the funeral and the start of the Jewish sabbath at sundown, no plans were announced for any diplomacy.
Mr Obama and Mr Netanyahu, who have had a testy relationship, last held talks on 21 September in New York, on the sidelines of the UN General Assembly.
Mr Peres will be buried in a Jewish religious ceremony, in a plot between two former prime ministers, Yitzhak Rabin and Yitzhak Shamir.
Rabin was assassinated by an ultranationalist Israeli in 1995 over the interim peace deals that he and Mr Peres reached with the late Palestinian leader Yasser Arafat.
"A light has gone out," Mr Obama said in a statement after Mr Peres died in a hospital near Tel Aviv on Wednesday, two weeks after suffering a stroke.
Outside Israel's parliament yesterday, an estimated 50,000 Israelis filed past Mr Peres's flag-draped coffin as it lay in state.
Former US president Bill Clinton, arriving in Israel a day before the burial, visited the plaza in front of parliament to stand, head bowed, in front of Mr Peres's casket.
The leaders of Egypt and Jordan, the only Arab states to have signed peace treaties with Israel, were not on the roster of participants issued by Israel's Foreign Ministry.
But the Egyptian foreign minister was scheduled to attend and King Abdullah of Jordan sent a telegram of condolences.
Mr Obama was leading a delegation of 33 US officials, including Secretary of State John Kerry and Nancy Pelosi, Democratic leader in the House of Representatives.
Britain's Prince Charles, French President Francois Hollande, Italian Prime Minister Matteo Renzi, Polish President Andrzej Duda, Canadian Prime Minister Justin Trudeau and former British leaders David Cameron and Tony Blair were also among along list of foreign dignitaries attending the funeral.
Israel laid on heavy security, deploying about 8,000 police and intermittently closed the main highway between Tel Aviv's airport and Jerusalem for Mr Obama's motorcade.
