The EU foreign policy chief, Javier Solana, said the crisis represented a failure of diplomacy and everyone had to work hard to end the suffering and advance the peace process.
In other developments:
Israeli President Shimon Peres says his country does not intend to occupy Gaza or crush Hamas, but "to crush terror". "We shall not accept the idea that Hamas will continue to fire and we shall declare a ceasefire," he says
In the West Bank, there are angry protests against the Israeli offensive. A Palestinian man is shot dead by Israeli troops who confronted youths near the town of Qalqilya, medical sources say
At least 17 people are killed and 130 injured when Israeli shells fall near a school and the main market in Gaza City, Palestinian health ministry officials tell the BBC
Hamas-run media report the capture of two Israeli soldiers, but the Israeli army says it has no knowledge of the claim
At least 32 missiles are fired into Israel from Gaza, targeting communities close to the border. Two people are lightly wounded in the Eshkol region, while one woman is slightly injured in Sderot
A tank shell fired in northern Gaza kills 12 people, most of them civilians, Palestinian sources say
An Israeli missile hits a house in Shujaiya, killing a mother and four of her children, officials at al-Shifa hospital tell the BBC
A senior Hamas military commander, Jihad Hamdan, is critically wounded in an air strike in the southern town of Khan Younis
Many claims cannot be verified. Israel is refusing to let international journalists into Gaza, despite a Supreme Court ruling to allow a limited number of reporters to enter the territory.
Witnesses say the attack began when Israeli military convoys supported by attack helicopters crossed into northern Gaza at four separate points after nightfall on Saturday.
By late Sunday afternoon, large plumes of smoke could be seen rising over the northern part of the Gaza Strip as Israeli forces stepped up their offensive.
The BBC's Mike Sergeant on the border with Israel says that although the sounds of explosions, gunfire and helicopters remain constant, the fighting seems to be moving away from the northern end of the territory, and heading further west.
Israeli tanks and heavy armour appear to have moved in on either side of Gaza City, in effect cutting the territory into two from the Karni crossing to the sea, he adds. The town of Beit Hanoun has also reportedly been surrounded.
According to Hamas officials and witnesses, the main fighting is centred on four areas: east of the Jabaliya refugee camp; in the Zeitoun neighbourhood to the east of Gaza City; on the coastal road close to the site of the former Jewish settlement of Netzarim, south of Gaza City; and in an uninhabited area in the centre of Gaza.
The Israeli military says Hamas fighters are not engaging them in close combat. Instead, they are using mortars and improvised bombs.
Palestinian medical officials inside Gaza say at least 35 people have been killed in the fighting since Saturday evening, several of them civilians, and dozens were injured.
The Israeli army said one of its soldiers had been killed and 32 wounded in the ground offensive, at least two of them seriously.
An Israeli military source told the BBC it was unrealistic to stop all the rocket fire, which has killed four Israelis in the past nine days, but the army believes it can be decreased and made less accurate.
The move into Gaza had been preceded by some of the heaviest Israeli air and artillery attacks on the territory in more than a week of bombardment.
The UN has warned of a worsening humanitarian crisis, and believes a quarter of the 500 Palestinians killed by Israel so far were civilians. Israel says about 80% of those killed were Hamas militants. A further 2,500 Palestinians have been wounded.
On Sunday, Palestinian Health Minister Dr Fathi Abumoghli said the Israeli military was restricting the movement of ambulances, leading to many casualties dying before they could receive treatment. One physician and two healthcare workers had been killed, he said.
The UN relief agency in Gaza, Unrwa, meanwhile said that about 1,200 people whose homes had been damaged or were in areas hit by fighting had fled to shelters it had set up.
Unrwa, which provides food rations for some 750,000 people, said it would continue distribution despite the fighting, but that its stocks of flour would run out in three days.
The Israeli authorities have co-ordinated deliveries of aid since the offensive began, but agencies say they have been insufficient, and that it is difficult to get supplies to where they are most needed.
The main aid crossing point at Rafah has also been closed while Israel troops attempt to destroy tunnels under the border with Egypt.
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